Cassette Deck doesn't auto reverse

This is a common problem....with tapes.....in the old days there was a shiny end of tape (EOT) sticker placed on the tape end which was detected as the tape end and caused the electronics to reverse the motor drive to play in the teverse direction.

If consumers alter the length of the tape by cutting and joining they used to cut off the EOT sticker and then the tape wouldnt reverse like in your case.

Now days a hi frequency signal is recorded on the tape and a detector circuit inside sees this signal and reverses the tape for you,

I would suggest that the simplest thing to do is to clean the playback head with a cotton bud and some special head cleaning alcohol.
There jusy may be a build up of magnetic material on the play back head which is stopping the high frequency signal getting through to the detector circuitry.

Different manufacturers do this EOT detection in different ways but the head clean is the first line of attack that you can do yourself.

Please turn off the power before doing this activity and be very careful with the cotton bud that you are not rough and dont push too hard on the recording/play back head.. Be gentle...
The cleaning liquid must be proper head cleaning liquid, so dont try shorth cuts and use just anything to do the cleaning or you may ruin the detector head surface.

The quality of play back sound will also improve from a good clean..

Cheers From YUBEUT