Nothing keeping you from doing it. It'd probably be best to go off the wires for the backbox GI. You can get the standard happ lighted buttons (https://na.suzohapp.com/products/pushbuttons/75-0004-1221044), and just use those. You'll have to drill your button hole a little bigger, and cut the two anchoring tabs off the button. A little woodworking, but hardly a killer project.

P.S. you can still get atari cone buttons, but they have teeeeeny tiny little lugs to solder to. Easier to use the bigger ones.

The button is countersunk only for aesthetic appeal. It'll still work just fine.

The one I linked to will work, as will any marked "Lighted button" or "IPB" (that's illuminated push button). If it's clear or translucent, it's probably a lighted button.