I worked on garage doors in the early 70's, before I got into the alarm industry, which used much the same remote signal technology for its wireless sensors as garage door remotes. As well as I can recall off the top of my head, that generation of RF receiver activated a relay if it received any strong signal at the right frequency, and the frequency was fine-tuned with a variable air capacitor; which was how you kept your remote from opening every other RF-equipped garage door on the block (and hopefully, vice versa).
RF receivers and remotes were a lot less common back then, and the environment was less laden with background RF, but we still had doors opening "spontaneously" from passing police cars and even airplanes who happened to broadcast on the right frequency. I'm frankly surprised that your vintage opener "works well", which I would take to mean it doesn't open spontaneously too often.
If you can find what frequency your receiver uses, it's possible a contemporary remote that transmits on that freq will activate it. Unlike the current generation of receivers, yours won't insist on hearing an electronic serial number (ESN) in a specific format before it activates its relay.
However, I have to agree with Gyrd3 that your old technology is a security risk.