Tape player
The most common cause of tape player failure is the cassette, not the player. Have you tried other tapes? Yes, you tried a known good one but perhaps not known good in that tape player. Try others.
If it is the player, it's usually the rubber pinch roller. Don't try to clean it with a normal swab, the hairs will get into the mechanism. Radio shack used to sell head cleaning swabs that had a foam or chamois cover - that's what you want. The recommended cleaning fluid is isopropyl alcohol. Methanol (stove alcohol) works best but is hazardous to your health.
C60 and C90 tapes have thicker base than 120's and will play much more reliably in a car. Pre-recorded tapes are frequently 120 base (thinner) even though they are shorter play length. So home-made tapes are more reliable so long as you use C60 stock.
If you really value a poorly performing tape and can not replace it, fast forwarding and rewinding it may get a few more plays out of it. But you're much better off recording CD's onto C60's and using those tapes in your car player. Also, never put a tape into the player until the player has a chance to temperature equilibrate (cool off in the summer or warm up in the winter).
Yea, I used to work in a tape plant. Could go on for hours.
Hope this helps, good luck
PS My '95 tape player had a dirty pinch roller when I bought it. Works fine now.