Stupid question... Was the FM actually working before you tried to
connect
the extra speakers? That is, did you actually hear the FM playing?

Yes, that's what confused me into thinking that something else was
broken. For a short while, the receiver was playing with the antenna.
But after a few minutes I tried to hook up the second pair of speakers
to the same terminals and the moment I did that it suddenly stopped
playing, as if some kind of fuse broke or some kind of protective
circuitery jumped in.

But for what I can read here (and I did not yet say this enough, but I
do appreciate all the help you have given, folks!) , both fm and
speaker connection would have nothing to do with each other...
Well, I think I must accept the fact that indeed something "down the
line" is broken and perhaps it's time to get it replaced with another
tuner.

Let me throw a little Latin at you: "Post hoc, ergo procter hoct". This is a
fallacy in reasoning that means "It followed, therefore it was caused by."

Barring some Truly Weird behavior in the receiver, this is coincidence. You
need to take the unit to someone who can properly signal-trace the tuner and
amplifier, and find the source of the problem.