After a lot of thought and playing around with different keypads, I think I'm going to go with low-profile mechanical keys. 

The other options:

• Standard electronic push buttons. I think the clickiness would be annoying, especially given how hard you have to press.
• Soft press electronic push buttons. These are nicer but would still need custom caps to make the area feel connected. This is something I dislike about most newer intellivision controllers. I want a big square button that butts up next to the other ones.
• Membrane keypads. This is what I originally planned on using, but they are all underwhelming and don't have enough travel in the click to feel accurate. You'd never be sure you pushed it.
• Touch screen. Well, this is an entirely different animal that clearly has great possibilities, as demonstrated on the DS version (and homebrew DS version). Also, a recent custom INTV controller with a touchscreen was put on github, which is really well done: https://github.com/rodineyhm/IntvTouch. However, I want the tactile feel to distinguish buttons.

I don't really like any of these options. So, my current plan is to use Gateron low-profile switches (https://www.gateron.co/products/gateron-low-profile-mechanical-switch-set) with some flat or custom keycaps. Probably something like the DOYS keycaps that have a raised circle (https://deadline.space/products/doys-keycaps). This would keep the overlays flat, but still let you feel the individual buttons before pressing, much like the original. I'll resin 3d print a bunch to see what feels the best, and it'll be nice for anyone to change the keycaps to what feels best for them.

Here's a quick picture of the keypad board. The additional wiring and resistors is for mapping the columns to use 3 analog pins rather than 7 digital ones on the Arduino.

The next steps are to get the keys wired up on a prototype board, start making some pcbs for testing with my cnc, and start playing with keycap shapes.

 Happy new years!