A Teensy 3.x - so a 3.2, 3.5 or 3.6 - is easily powerful enough to do the things you want. If you know it and like it, that’s good. They’re not too expensive, nice and small, and easy to integrate into other circuit boards.
With your button board, you might want to look into 4067 multiplexers (eg the 74HC4067), or traditional keypad multiplexing routines (where you use one digital output per row and one digital input per column). Basically, anything you can to reduce inputs.
Mechaduinos: these seem pretty heavyweight, in that it’s a nice servo motor, plus motor controller (which are always a faff)… but they’ve each got a Cortex M0 on as well? Something in me wants to run everything from the same microcontroller - but it might make things simpler to use what you know. Those servos also look quite… hefty (both in terms of weight, and thus power requirements).
Encoders are not a problem, they need two data lines per encoder, there are librariers for them.
Touch inputs might need some more detail, especially if they’re capacitive.