If you can find a cassette cleaner cartridge, you might try applying a little Isopropyl alcohol to the fiber tape in the carteidge and running it throygh the player. This not only cleans the play-back heads, but it also helps to clean the capstains/rollers and pinch-wheels.
I know this isn't encouraging, but I think that road dust/grime is another reason why the c.d. players seem to be a little more reliable. If only they would skip less...
I also think that the original rubber drive belts are most likely dried out and/or worn.
If you're comfrotable with tearing the radio down, do so, and remove the belts, go to a stereo repair shop and find some new drive belts, install them, lube the bushings/bearings with light oiland clean the heads carefully with alcohol, reassemble the radio and if all the electronics are good it should work.