Whilst it's usual to name a D major chord (in key C) as V/V - the dominant of the dominant - is it legitimate to call Bb (b7) IV/IV - the subdominant of the subdominant? Another justification is of course that it belongs to the parallel key (C minor), but this would be Roman numeral analysis.
As is often the case for harmonic analysis: context is key. Especially true I feel when looking at postwar popular music, in which harmonic conventions vary a lot between period, genre and composer, and so a "one size fits all" approach to functional analysis is even less reliable than in common practice music.
A general principle though is that plagal resolution and I IV interplay is the bread and butter of "classic" rock, and of rhythm and blues, and arguably more important than dominant to tonic resolution. However, the complexities of tonality, modality, and functional harmony often give ambiguity to how chords function within this framework.
When is a ♭VII a IV/IV?
When it isn't:
1 ♭7 5
Well, let's start with where it isn't. Here is an example you gave of C > B♭ > G. For some reason the song that jumps into my head is an obscure album track by Booker T. There are obviously millions of (more well known) examples of this, but this will do fine.
This is in B, and it features a prominent B > A > F#. Another example would be "fire" by jimi hendrix, in the bridge, you have D C A C
Here, the A is certainly a ♭VII chord, straight up. This is sort of chord is "what it says in the tin". The ♭VII is a ♭VII chord as much as a V chord is a V chord, it's certainly not borrowing from the IV: it doesn't even involve it. Now, how to analyse progressions like E G A or E D B functionally is another interesting question (for an example of this, try and analyse where there is or isn't a modulation in "my sharona"), but in any case, something it certainly has nothing to do with is plagal cadences, so you can rule out /IV.
♭7 > 1
Another instance where it absolutely isn't a IV/IV is where it's used for its (weaker than both IV and V) pull to the tonic, that is to say when ♭VII is used as a pseudo-dominant to I. In this regard, using ♭VII to lead to I might be viewed as analogous to IV V in dominant preparation, while a ii or II are a stronger pull to a V, the IV also has a pull upwards by a tone. Use of a ♭VII > I exploits the same property to resolve up by a tone. This usage is common in jazz and Anglophone folk music.
This can be taken a step further with the ♭VI ♭VII I progression (aka the "Billy Shears" Cadence), where the "up by a tone" progression is extended further (you could think of it almost like a secondary subtonic, analagously to a secondary dominant)
In jazz this is sometimes called the "backdoor progression", and is more commonly voiced as iv7 > ♭VII7 > I, which has better possibility for more "contrapuntal" voice leading and chord extensions than a simple ♭VI>♭VII>♭I, which is all parallel motion and would usually stick out like a sore thumb as a resolution in a jazz tune.
When it is:
Now take a sort of "Obvious" case from the other side: Free Bird by lynrd skynrd. In this case, the F chord leading to the C chord is absolutely a modulation, a "double plagal cadence", it's completely unambiguous. Before the chord comes, you're definitively in a "G major" space (in the strictest, #7 sense), and the F is a modulation. In that sense, the F chord is kind of the plagal equivalent to the common classical device of briefly modulating to the dominant for a couple of bars through sharpening the fourth degree. (as in "by the dawn's ear -LY light" in the American National anthem, where the "ly" tips you off to the upcoming dominant chord.
Now in other instances, it's far less obvious/unambiguous than these 2, and you could probably make a valid argument for either noatation.
So far my examples have served to show either 1) absolutely no "secondary plagal" characteristics at all, and 2) a situation where the Ionian tonaility of the song proper is so unambiguous that the introduction of the flattened 7th feels definitively like modulation (i.e. ♭VII is clearly functioning as IV/IV)
Ambiguity: Contrasting 7 4 1 with 2 5 1
Outside of these more obvious examples, there are many contexts where whether you have a IV to IV or a ♭VII becomes debatable. By analogy, let's look at a 2 5 1.
Take the chorus of "I wanna be like you" (or a number of standards), you'll find something like:
| C | A7 | D7 | G7 |
Most people would view this as a series of dominants of dominants NOT as I VI II V
But, if you take a doo wop progression:
| C | Am | Dm | G |
Then no one in their right mind would consider this as anything other than | I | vi | ii | V |. Despite the fact that there is undeniably a pull fifthwise from the Am to the Dm to the G, the lack of a leading tone/major third leads us to, rightly, not view this as a modulation.
There is an easy indicator: third degree getting sharpened tells you exactly what's going on. The reason there's not much ambiguity here is that you're rarely if ever going to find a song in C that includes the above progression where an F# or C# could reasonable be viewed as just modal borrowings: for the #4 as a borrowing it would have to be lydian (unlikely to say the least) And for the A chord, the idea of an augmented tonic makes no sense at all, and it's clear that the C# is not a Db (this after all is not a B double flat chord!).
Contrasting this to 7 4 1, there often isn't the same clear indicator for the situation of IV/IV vs ♭VII chord. Why? Because while you're unlikely to find a tune where II V I can be explained away modally, often the flat 7 as a note can already be considered "fair game" without a modulation. It’s perfectly conceivable to have a rock song in G: where an F note is not really a borrowing at all, nor is it a blue note*. For want of a better word, you have plenty of "mixolydian feel" rock songs (albeit with blue thirds and fifths in the melody). Think "you really got me going" for the archetypal example of this. Even though we're using the classical analytical technique of roman numeral analysis, it's necessary to get out of the idea that a major scale is sort of the "default" for a song with a major tonic chord. Similarly useless is the idea of a juxtaposition between EITHER "modal" or "tonal" **.
So there are plenty of examples of where, while the ♭VII does indeed serve to pull to the IV chord, it still isn't a modulation. If the ♭7 scale degree is already "in key", then whether you have a ♭VII or a IV/Iv is more ambiguous (analagously, it would be like trying to decide between V/V and II in a lydian context).
The outro to Hey Jude is an excellent example. F > Eb > Bb is the chord progression here. There is absolutely no subjective sense of "modulation" between any of these 3 chords, and so rendering this as |F| IV/IV | IV| would be obtuse at best. The key centre is so strongly "mixolydian" that if you were to improvise melodically over it, even over the F and the Bb chord, an E natural would sound like a chromatic passing note to either Eb or F, not an "in key" note.
So we have a contrast between free bird and Hey Jude, while both songs use 7 -> 4 -> 1 , in free bird this feels like a modulation (as the 7ths is "out of key as a melodic note in the context of
I) whereas in Hey Jude it does not (as the 7th is idiomatic melodically speaking to that whole section of the music) In reality, the majority of cases show more ambiguity than these more clearly defined cases.
Take, the who, won't get fooled again:
A G D A G D We'll be fighting in the streets, with our children at our feet, A G D E and the morals that they worship will be gone.
In the verses, you've got a strong mixolydian element, the G note doesn't feel like an accidental, and so it seems overkill to consider the G chord in the verses a borrowed chord every time. But then, nevertheless, at the end of the each couplet, you land on an E (MAJOR) chord. In a sense, it's almost accurate to view the song as being "in" A mixolydian, the G chord as a natural element of that, and the E chord as a borrowing from A MAJOR. But obviously, that's a bit ridiculous too. What's more accurate is just to say that the interplay between "mixolydian" and major within rock music is often something in constant flux. If it helps, you can view the way harmonic, melodic, and natural minor function in classical music to the way "blues-rock-y" tonality functions in rock music. A mix of both flat and sharp 7ths is seen as par for the course in common practice minor music: you need simply extend that to major too.
A final thing to note from the same song, is that, after the E chord, you get an interesting turnaround: C > G. A lazy theorist might describe this as "non-functional" harmony: but it isn’t really, if you're willing to accept what the functions are. The C functions here as, brace yourself, a secondary plagal to a bVII chord. The G is acting as the ♭VII chord (a subtonic with a pull to I), and the C is its "secondary plagal".
So what we have here is
| E | C G | A | | V | IV/bVII bVII| I |
*Which is a separate concept altogether, and cannot be used in a hand wavey fashion to "explain away" any example of complex modal interplay within blues influenced forms. While a flat 7 may be a "blue note" it functions as such only when used a) as a tension against the tonic or b) as a tension against the third of the V chord. A flat 7 as a consonant melodic interval shouldn't be misconceived as a blue note any more than a straight (uninflected) minor third should be considered a blue note within a minor key.
**Or at least, if the distinction is to be maintained, it must be separated from the observation of whether a song "uses modes". Something that may use a "mode" as its home key can nevertheless be functional and, conversely, one easily can write non-functional "modal" music using the only the ionian mode.
Call it IV/IV if that is what it IS in context. But also be open to the idea that functional analysis isn't always appropriate or useful.
You won't like it, but it depends on context (i.e. how musical material is prepared / organized and / or listener perception). Music can be heard / interpreted in a variety of ways, and thus, how that music is analyzed may also be varied. Someone may in fact hear IV/IV; others may not. Theoretical debates ensue.
Roman Numeral Analysis is just one method of analyzing music, but it used to show chords' harmonic function. Labeling secondary dominants as "V/V's" shows the function of a non-diatonic chord relative to the parent key. In other words, the analysis reveals how tonality is established and thus, how the two keys (or tonicization) are related.
A chord based upon the flatted-seventh scale degree is also non-diatonic, but that chord does not purport the same harmonic function. In this scenario you describe (IV/IV) we are being asked to hear a subdominant chord in the context of another subdominant chord. A move from the subdominant to tonic is much weaker than a move from dominant to tonic; the fact that both subdominant and tonic share a note inherently weakens the harmonic pull from one chord to another.
Remember that analysis isn't based upon what the music looks like, but rather, how it is heard. Hearing, after all, is what matters the most.
But for argument's sake, let's say we followed the progression: I, "IV/IV", IV
Depending on context, after the arrival at IV, you wouldn't feel as though you've changed keys. You would just feel like you've reached the IV chord of your parent / current key; the bVII chord in this instance purporting more of a pentatonic / mixolydian feel than a pivot chord for a key change.
A much more striking (and common) way to modulate to IV would be to take that same b7 and throw it on the bottom of your tonic chord, so it makes a V4/2 / IV6, resolving to first-inversion. The reason here is that the interplay between the b7 and the 3rd of the tonic chord create a tritone that resolves by step in contrary motion.
With a bVII chord, the tritone is replaced with a perfect-fifth, a stable interval but the 2nd least harmonically-active. Add to that the fact that in order to execute this progression, it requires parallel motion between chords, which is the weakest chord motion of the types (parallel, similar, oblique, contrary).
Okay smart guy, so what would you call it, then?
Depends on context. If the music is not functionally-tonal, then it wouldn't be the best idea to use RNA (why would you use a hammer to cut a piece of wood? Wrong tool for the job). In that situation it would likely be more helpful to use Macro Analysis, Set Theory, a matrix, or even Schenkerian Analysis. Again, would be difficult to know without context. The larger point here is that you can't just shoe-horn any analysis model into a piece in order to justify your viewpoint or knowledge-set.
Thinking more broadly:
Irrespective of how a piece is analyzed, it is important to ask does my analysis reflect the way the music is heard? The musical frameworks and models you use that most fully substantiates a confident response to this question present you with the most likely to-be-accepted analysis.
Hope that helps.