The problem I'm having is caused by the capstan sinking into the recorder, due to a missing e-clip, a.k.a. retaining washer, that's supposed to hold it in place. I ended up finding the post that describes the same problem (
https://homerecording.com/bbs/user-forums-brand/tascam-user-forum/414-mkii-capstan-slipping-364003/). Using the info gathered there, I took a pair of needle nose pliers and gently pulled the capstan shaft up into it's normal position. Holding it in place with my index finger, I then pressed play. As the pinch roller and capstan met, the takeup reel began to spin, but not for long as the capstan fell back into the machine. Messing around with this for a minute, I took a break and looked at my Porta 02. Sure enough, there are a few extra clips on its capstan that are absent from the 414's. Over time, with regular use, I believe it's possible for these clips to break or fall off.
To anyone that ends up with the same symptoms, I would advise you to check this first, as you don't have to open up the machine. If you look at the capstan and there are no weird "C" shaped clips attached to it, and you can lift the shaft up with your fingers, or GENTLY with a pair of pliers, this could be the issue. If you have no idea what the clips should look like, do a search for "capstan e-clip" or "capstan retaining washer". I hope this helps somebody in the future!