i had similar issue with mine but the radio kept draining the battery because i had linked the switched live with the permanent live, so you could have the radio on all the time.
in the end i discarded all the existing wiring and put all new in and it worked much better.
there are plenty of switched lives to be found in the fuse box, just check out the faqs section of the forum.
this way when you put the key in the radio will come on, take the key out the radio will go off.
problem with this old cars is the wiring for the stereo often gets abused by previous owners messing with stuff they dont understand so you end up with a rats nest of different colored wires and scotch-locks
plus all that cable will be old especially the main live feed, i fitted an alpine stereo to mine and it used to cut out with loud music so i had to up rate the cable.
now its banging