Yes, thin would be super hard to do - I expect at least a good few CM in depth. The Alps fader alone is almost the biggest factor.
This is a link to a German electronics store that offers the same I have been using: https://www.reichelt.com/de/en/alps-rsa0n-studio-fader-motor...
I will see if i can find the box with my prototype setup because its way smoother than your setup is. its unbelievable smooth when using a cheap analog servo board.
From your video you came to almost exact the same solution I have. :)
I really can suggest trying to run it from a servo board - it has all the stuff for smooth movement already in it.
I'm using a Pi Pico now with a stepper motor (no video of it yet), and it's super smooth, indeed!
I was thinking about using a "SoftPot Membrane Potentiometer" to bring the chonkiness down a little, where the knob on the front would have a little nub at the back which would press against the strip and give you the same kind of effect (but then you need a way to drive that nub, of course).
I also thought about using the stepper motor and spiral combo out of a CD-ROM drive, maybe putting an encoder on it, but the spiral isn't long enough, so meh.
I'm still waiting for screens from AliExpress before I make an updated video, otherwise it'd just be, "look, it's smoother now, yay".
From your video you came to almost exact the same solution I have. :) I really can suggest trying to run it from a servo board - it has all the stuff for smooth movement already in it.