Is this an effective way to think of and practice working with chord
extensions?
If you are able to see chords as combinations of other chords, it can help you play just the right notes that make the difference and add the right taste. The 9 chord being a combination of major + minor triads is a common example - if the rest of the band is playing a plain C major, you can make it a C9 by playing a G minor or Gm6. And if you know that Gm6 is equivalent to Em7-5 (just with a different bass note), you have lots of options all over the fretboard for building the C9 chord. Actually, Gm6/C has all the same notes as C9.
"Where's the nearest full C9?"
What fingerings do we have for a full C9 chord? This is probably the most commonly used one:

Or if you want something different, you might put together something like this:

I don't particularly like the sound of that option. But anyway, to sum it up, all two of your options are here:

If those are the only ways you know for handling a C9, you may feel a bit uncomfortable and restricted. There must be a better way, this can't be how guitarists have to do it?!
Well OK there's this third alternative fingering I just discovered, but haven't used:

"Where are the nearest components of a C9?"
It's almost never necessary to play all of the theoretical notes of a chord. If you just want to add the juicy notes of a C9 in a solo, or play C9 flavor with fewer notes, and if the bass is already done somehow, for example if there's a bass instrument, or if you just played it a second ago and it's still echoing in the harmonic context, why not dissect the chord and play a Gm, Gm6 or Em7-5 instead. Now we have many more options all over the fretboard. All sorts of different ways to do a C9. (there are probably errors and omissions in the animation, but you get the idea)

Try strumming or arpeggiating one of these Gm6 or Em7-5 chords:

If those shapes don't look familiar, here are the corresponding plain G minor chords:

Learn the generic pattern, not all cases separately
However I find this quite daunting as the sheer number of
possibilities across all keys is a little overwhelming.
Here's where your problem might be. First of all, you don't learn all combinations in all keys separately as hundreds of special cases. You learn the generic pattern relative to the chord's root and relative to the tonic. To make a 9 chord, you play a minor triad or a minor sixth rooted a fourth below or a fifth above the original chord's root. From C you go --> Gm6. From A you go --> Em6. The trick is to keep track of the important strategic locations. Where is the tonic note? Where's the relative major/minor's tonic note? Where are the third and seventh of the currently playing chord? Everything that happens, is relative to these strategic locations. You learn the generic pattern, not each and every possible instance and variation of it.
Here is one generic pattern for seeing the minor triad over something with the root note on the low E string. Notice that it doesn't say which frets these are. The pattern is the same all over.

You learn the tricks one at a time
And secondly, you don't try to learn them all in one step. You eat the elephant one little piece at a time. Slowly. First you learn and practice the generic pattern for one chord trick, in one familiar song, in one familiar key. Just ONE. And when you get that one small step done, you are happy, because you achieved something. Then you take another small step, but only after you've taken the first step. Do not watch Youtube videos, they just distract your focus away from taking small steps. I don't personally know any great jazz players, but I'm sure they all learned everything they know one step at a time.
Some other chord substitution tricks where you can superimpose different chords relative to a chord's root note to make a larger chord:
- A C maj9 chord can be seen as a G major chord over a C major chord, or as G6/C or Em7/C. If the bass player is playing a C, just find any G major or G6 or Em7 chord anywhere on the fretboard, to add the money notes.
- An 11 chord is a major or major six chord rooted a whole-step below the root (i.e. on the seventh). To do C11 you play Bb or Bb6 over a C bass note. This is used like a Csus4 or C7sus4 chord. (Actually you can even do Bbmaj7/C for the same thing, even though there's an A note in there.)
- On a dominant seventh chord, play a dim or dim7 chord rooted on the dominant seventh chord's third or its seventh, or one half-step above its root (which have the same notes). Practice this in songs where there's a dominant seventh chord doing a V - I motion. (which should be nearly every song ever written)
- On a dominant seventh chord, prepare the listener for the strong dominant by first making it an 11 chord, then a 9, and then the dim7 trick explained above
- On a dominant seventh chord, play major triads from the half-whole diminished scale of the dominant's root. For example on a written C7 and bass note C, arpeggiate A major, F# major, Eb major triads. (if you add together the notes of those triads and the bass note C, you get the C half-whole diminished scale)
By the way, this chord-combination thing can be used for thinking about modes as well. For example in "regular" A minor you have A minor and D minor chords, but in A Dorian you have A minor and D major. In regular C major you have C major and D minor, but in C Lydian you have C major and D major.