Just wanted to update on my issue in case anyone else here experiences this problem at some point.
After being in touch with Tascam themselves and various technicians across the US, one of the techs pointed out to me that there should be a slight cut in the capstan shaft, where a small plastic or nylon washer (an e-clip or similar would likely be too thick) should be present to hold it in place. And sure enough, when the capstan is in the proper position, the cut sits right above where it comes up from below the deck.
Of course, after consulting their service manual, Tascam’s parts department was like “yeah sorry this part is discontinued,” and because of how thin the cut is (less than 0.5 mm), it looked like I had very few (if any) options beyond custom ordering a huge batch of washers.
Obviously not wanting to do that, I started looking for other solutions in true DIY spirit. Since the part needed was made from plastic or nylon, and as a guitarist I have tons of guitar picks laying around, I ventured out to the hardware store to grab a drill and some drill bits.
I came back with a hand drill and a smallest (read cheapest) set of bits I could get with one that was 1.5 mm diameter (approx the diameter of the capstan shaft). Then 15 minutes later, after using a box cutter to hack away a small piece of pick and filing the hole ever so slightly bigger than 1.5 mm, I slid my DIY part on, and to my amazement it actually worked!!!
Honestly I didn’t really expect it to work. Apparently neither did my partner, when after I showed it to her said she was honestly afraid it wouldn’t work and that I would’ve just gotten more frustrated (pretty accurate tbh).
For reference, while the pick itself isn’t labeled as to what size it is (it’s an Apogee promo pick by Ernie Ball), I suspect it’s around 0.5 - 0.7 mm thick.
Needless to say, I’m so glad to have a functioning 4 track now!!