Some thoughts about game console controllers:
(1) They are designed to be as cheap as possible to mass produce, much like the keyboards that come with $400 desktop PCs. As a result, any feature which doesn’t have obvious marketing value is going to be cut. Most relevant here, the buttons on game controllers are really crappy little rubber domes which are in general pretty unpleasant, too stiff with poor feedback, easy to wear down / break, and usually requiring a pretty good mash to consistently actuate. However, you could definitely use better buttons for this kind of project, whether keyboard switches or some kind of tactile switches or other micro switches. There’s a whole world out there of nice switches of various types to explore.
(2) They are designed to be pressed with the hands and arms in a very specific orientation. If you having your hands in a typing position, I think you’ll find it somewhat tough to find a good position and orientation for a standard game controller button pad to be comfortable to press all day long, especially if you want to use it as a modifier for finger keys.
Still though, having several smaller thumb buttons oriented out of the plane of the rest of the keyboard is a reasonable idea.
Definitely try to build a prototype and let us know how it goes!