I don't think the songs in question really have circle-of-fifths progressions like you claim.
First example song
The chord "cycle" in the first one starts with the D, not A or A7. If you see the A-based chord as the first bar, then you have it messed up. The A is the dog's tail, not its head.
The D - E chord pair is a thing in and of itself. You're supposed to hear it as potentially being a part of a D - E - A - A progression fully in major. But instead, it takes a "surprising" (in anime pop this "surprise" is such a cliché that it's an elemental part of the genre) turn to the minor side, ta-da! you have C#m - F#m instead of A - A. Who would have guessed. (Both triads have two notes in common with the expected A - A, by the way.) After that it takes a second round along the same lines: Bm - E - A - A, where the Bm is really the same as D, but from the relative minor side. You could do the entire progression as D - E - A - A - D - E - A - A to be fully on the major side. To do the same fully on the minor side is: Bm - C#m - F#m - F#m - Bm - C#m - F#m - F#m. Or even Bm - C#7 - F#m - F#m - Bm - C#7 - F#m - F#m. Try it, sing the melody on top of those chords! Even on a C#7 chord, an E melody note is fine, it makes it a bluesy C#7+9.
Second example song
(IMO the song doesn't really go like this, but assuming it was)
- Bb - C7/Bb - Am7 - Dm7 - Gm7 - C7 - F - (F)
This is the same thing! If we transform/simplify it fully to the major side of things, it's: Bb - C - F - F - Bb - C - F - F.
The Bb - C7/Bb chord pair is another anime pop cliché, it's a thing in itself. One of my favourite things actually - I'm always looking for a place where I could fit in an IV - IV/V pair, regardless of what happens before or after. ;) Nothing to do with circle of fifths.
Why is ii dim in minor common, but vii dim in major is not?
There's another question embedded in here though, and you explicate in your comment to Aaron's answer: "Why do they use ii dim in minor but rarely go with vii dim in major?"
Basic pop harmony works like this: there's a harmonic balance, and a center of balance called tonic. Home. The harmonic balance can lean on either side, let's call them dominant side on the left, and subdominant side on the right (to correspond to the layout on the piano keyboard, G is on the left side of C, and F is to the right). In addition to that, there's another dimension and that's a dark/bright dimension, meaning that the harmonic balance can be either on the major side or on the minor side. Ok? In other words, you can simplify or dumb down a pop chord progression to a I - IV - V version that does roughly the same thing. Or the same on the relative minor side. Here are the basic triads of C major and A minor shown in a chart of harmonic balance:
This maps six out of seven diatonic triads to some role on the harmonic balance chart. But it leaves out one diatonic triad, the vii dim in major, or ii dim in minor. Where does a Bdim chord tilt the harmonic balance? Is it a left-hand-side or right-hand-side chord? IMO, it depends on whether you consider C or Am as being the tonic. If you're expecting a C tonic, then Bdim is a left-hand-side chord. If you're expecting an Am tonic, then Bdim is a right-hand-side chord!
So, why is Bdim not used so often as a "V chord" in C major? Maybe for several reasons, but at least because it doesn't give a proper V-I bass motion. A G7 or G works much better, because it feels like a complete balance shift from left to center, AND there's the V - I bass movement. Bdim can do that, but it's not as strong.
But why is Bdim (or Bm7-5 or Dm6/B) used so often in A minor? It's because it does the job of a right-hand-side Dm chord very effectively (having D and F notes), BUT it also gives a very good V-I bass motion into E! And the B note makes the D and F feel somehow more bitter or sour, like a Dm6.
The important thing to realize is, many perspectives into understanding and reasoning about harmony are true at the same time! Chords are not monolithic objects - you can morph a chord to another chord one note at a time, and it retains some properties of the original, while dropping some and adding some. For example, you can drop the C from a C major and add a B instead - it still does part of the job of a C major, but not all. You can substitute a G major with ... a G#dim, and it will do part of the job of a G major. But not all.
If you can see even a partial circle-of-fifths progression somewhere, then you might be able to use some tricks you can do over a circle-of-fifths progression, but not all. Chords are not atomic or monolithic things, they are a combination of their parts.
If you can get to the next chord by a one-step motion or a step of a fourth up / fifth down in the bass, then it's a nice thing to have in any arrangement. For example in the second song, Bb - C/Bb - Am7 ... it's a nice step from C/Bb to Am7, because you can do that just by moving the bass down a half-step. And the C/Bb chord in itself has a slight Bb Lydian feel, which IMO is part of its charm.