First off, I'd like to point out the Circle of 5ths has been around for 100's of years, and falls under the category of "Theory". Theory, by its very nature, is fluid. It evolves over time and is subject to changing interpretations (see: Flat Earth Theory). It is also subject to continuing discoveries. There are relationships within the Circle that are not addressed in any of these 5 videos and likely some as yet undiscovered.
The Circle of 5ths, when applied to Stradella, is commonly seen as a "roadmap" to chords and their progressions. The I, IV, V progression, being major chords, is the basis of the Stradella system. The placement of the IV and V chords, surrounding the I bass, facilitates the fingering and reduces the required hand movement necessary to accomplish this common, rudimentary bass/chord progression. Moving up or down the fundamental bass row, to re-root the Key, follows the Circle of 5ths.
On the treble side, CBA (5-row) accomplishes a similar "convenience" by means of a duplicated finger pattern, in order to achieve the same three chords (I, IV, V). When playing in any Key, the pattern remains the same. Applying this phenomenon to, say a PA, the fingering will be different for most every chord. The same applies to scales. The Circle of 5ths., while normally seen as a "dictate" for chords (I, ii, iii, IV, V, vi, vii), also dictates scale orders and their relationships (see vid 5 - 1:20).
The relationships between scales are the same. For instance, the relative minor for "C" is "Am", for "D" it is "Bm," "F" is "Dm" and so on. If you look at your button board, you'll notice that "A" is located at the adjacent button above (visually) the "C" button. The "B" button will be found just above (visually) the "D" button (C system, B is the opposite). Keys and their relative minors, are a fundamental part of the Circle. This relationship carries on throughout the CBA layout.
The Circle also defines the locations of the various intervals (see 3rd video - 9:10) that will repeat, pattern wise, over the button board.
Some will say "This is transposition", and it is. However, transposition, via the button board ,would not be possible if there were no relationship between the button board and the Circle of 5ths. Again, I would point to the PA as an example.
As a suggestion, refer to the last two vids and apply what the guy says to your button board and see if these relationships don't jump right out.
Press on....
Waldo