Most probably a wire to power the motor of a car antenna, which
automatically extends when you switch on the radio.

I agree, it's very likely there to trigger a power antenna. You could use
it with a relay to trigger an amp (you can't power the amp through it, of
course).

Also, there is no mention of connecting any wire from the radio
directly to the battery for a continious power source. The main lead
(red wire) is supposed to be connected to +12 switched. The old radio
(that I removed) had a wire connected directly to the battery with a
small plastic fuse casing in the middle. Do modern radios use a
different method to maintain memory when the power is switched off?

Yes, non-volatile memory chips.

But what about the clock? Internal battery (aka planned obsolescence)?