Garage door opener controlled via generic remote
The simplest way to fix this problem is to purchase a "universal garage door receiver" or "universal radio control replacement" which does as the asker proposes. It is a receiver box that wires into your door pushbutton circuit and pairs to a new remote.

It is also useful for modernizing fixed-code openers for security. However, I understand that the industry safety standard is that no opener without a optical beam may be retrofit or repaired; those must be replaced completely.