It's likely the capstan belt and probably not the motor.
By the sound of this clip, I'd say the belt is stretched and worn out beyond it's practical useful lifespan, and just before it gives up the ghost, it sounds like THAT--->.
This is relatively common in the cassette world, or for almost any other analog tape deck. The belts and rubber parts deteriorate with age, stretch, and sometimes even lose their consistency altogether and turn to goo, in extreme cases.
[You know,... make sure it's not the tape or some other no brainer, before tearing into the tape deck to repair.]
I've done cassette repairs, mostly on Tascams. I buy broken units and fix them, and I know a lot about the typical cassette mechanism. I think that the moderately skilled and committed DIY'er can replace the capstan belt, and even the other internal rubber parts, but be forewarned, it's a lot of small parts in disassembly of the cassette mech, and probably not for the faint hearted or first-time-DIY'er. BTW, I'm a hardware tech by trade, 20 years of takin' stuff apart, and I do this stuff a home sometimes for "fun", so it's no big deal,... but first-time-DIY'ers be forwarned.
First time DIY'ers may have a little trouble figuring out how to take a cassette deck all the way apart,... much less put it back together, especially a 4-tracker.
It can be done, and it can be "fun", and you can sure save a lot of money with DIY-repairs, and you should be okay if you disassemble things methodically, and keep track of all the small parts, noting which way things are supposed to go back together.
In short, it's likely the capstan belt. Belt replacement is not that heavy duty of a repair, as compared to the "idler", but that's an explanation for another day. Good luck in repair.
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Hey, let me not forget to say it's a very cool thing for any member of this board or any home recordist to give someone a 4-tracker. Not only that, but you're probably one of the few people on this board who's recording on a Vestax, one of those 'me-too-Tascam-clones'.
Anyway, 'dya ever see the old '80s Vestax that was a 6-channel/6-track/rackmount cassette version? Ooh, six, cool. That's gotta be a collector's item now. I guess 6-track cassette format never really caught on.
Further on a tangent, 'dya ever see the AKAI 12-track portastudio-style recorder that recorded onto large 1/4" cassettes, [like the "Elcaset" 1/4" cassette]???? Ooh, another collectors' item!
If you look hard enough, you may find these rare gems on Ebay, once in a great while.
Anyway, thanx for reading, and I'll let you have your thread back now.