Thanks again - this was a bit of a surprise!
If I may submit an extremely pedantic music nerd bug report: at 46s in the video demo (https://www.youtube.com/watch?v=qboig3a0YS0&t=46s), the display should read Bb instead of A#, as the key of C minor is written with flats :)
(The precise rule is that a diatonic scale must use each letter name for exactly one note, e.g. you can't have both G and G# in the same key, and you can't skip B. This has many important properties that make music easier to read and reason about, such as allowing written music to specify "all the E's, A's, and B's are flat" once at the start of the piece instead of having to clutter the page with redundant sharps or flats everywhere.)
Regarding flats and sharps: one could ignore the Pythagorean stuff and go full well-tempered dodecaphonic, thinking purely in terms of semitones in the intervals. This toy sort of nudges towards this. It would be fun to add 12 small LEDs along the faders, and show the number of semitones with them, relative to the previous fader's position.
On one hand, the fact that the same sound can be named A# and Bb may be puzzling for a kid (they could differ on a violin, I suppose); OTOH if the kid later learns formal music notation, this becomes helpful, so your comment holds.
I think that, given the toy is (currently) diatonic, and doesn't really have any ability to visualize the chromatic scale (like a piano keyboard does), using the formally correct note names is more intuitive. That way, only the accidentals change when you change modes ("when I change it to C minor, the B becomes a Bb"). This naturally teaches a simple and correct mental model: "the slider chooses a letter and pushing the orange knob makes letters flat or sharp".
If you only ever use sharps instead of sticking to the correct notation, then the notes change inconsistently between different keys ("changing from C major to C minor turns the B into an A#, but changing from C# major to C# minor changes the F into an E"). This is incomprehensible unless you've already memorized the piano keyboard layout.
The OP's choice of restricting to the diatonic scale seems sensible to me -- it helps the kid learn the vocabulary of Western music (if that's your goal!) and it benefits the parents as well by making it hard to create something that sounds bad.