And here comes this week update. After many tests, I've pinpointed the source of the noise.
The excellent news is that there's nothing wrong with the design of custom electronics boards, neither the control board, nor the sensor board.
The good news is that the LED constant-current power supply which I built (based on the linear voltage supply I purchased) is just fine as I thought/hoped.
The culprit is computer which I wanted to use to control everything. Even powering everything by batteries and connecting only the Ethernet cable to capture the ADC data injected the noise. Powering the LEDs with the linear constant-current supply, the piano with the batteries and connecting the prototype to a remote computer with a very long Ethernet cable (because I can't move the computer let alone the piano) shows no sign of that noise.
On the other hand, I now need to figure out how to power the main board from a different place than the USB, which probably mean from the same (or similar) power supply I already have (but if I use the same, that has a too high voltage, so I'd need to step down twice -- need to think about it more).
I also need to investigate if the same problem arises when connecting to the computer via MIDI rather than USB, since I had no time to run that test yet (I doubt it since @gzpiano tested the heck out of it, but it's not impossible since perhaps his chain is cleaner than mine). If necessary I need to nudge him to design some sort of filter to prevent this problem from happening. Also, I need to ping @vagfilm and @Akari -- guys you need to make sure you don't inadvertently run into it too, since you want to use the USB-MIDI rather than DIN-MIDI.
I guess this data is useful/interesting for @DavisB too, in the sense that powering from USB makes a lot of other challenges arise, as we were discussing in this other thread
And for completeness, I think also @xooorx @candela and @aleath -- guys, sorry if I just made noise in your inbox, but perhaps this is useful information for you too (just let me know if you prefer refraining me from pinging you like this in the future)