Many processors do have this and expose it to programmers, it’s called “register windows” (though usually used for procedure calls). Or you can have banked registers, which often serve different privilege levels. Once you start looking at the microarchitectural level, you’ll find that modern processors have large register files and rename them into the architecturally visible ones.

Registers are expensive, yeah, but pushing them onto the stack isn't the most expensive part of a context switch on a modern cpu anyway. Switching the page table, and blowing away the TLB is. Pushing all the registers is some nice sequential memory activity and the stack area is frequently accessed and unlikely to have contention, so it's easy to cache (other than one or the other of push or pop has to go backwards, so you better have predictive access in both directions)