First I would take the door off, and lay it on the flat surface and check everything make sure nothing is broken, worn, etc.

Check the bottom pivot on the door maybe it needs adjustment... clockwise to lower the door and counterclockwise to raise it and if you have broken or worn parts just get the bi-fold hardware kit it has all the parts and replace what needs to be replaced.

Sounds like a worn part or an issue with the hole (such as a split in the wood). As suggested above, inspection should show you what is malfunctioning, and whether it can be easily fixed or not.

Everything looks fine visually - no cracking or split wood anything like that. The only problem is that regardless of how high I raise the door by adjusting the bottom pivot, the moment I try to stand it up vertically, it sinks back down to the point where it's almost touching the ground. The bottom pivot is supposed to support the weight of the door... right?

So I got this kit from a local hardware store:

https://www.amazon.com/Prime-Line-Products-7071-Bi-Fold-Bracket/dp/B00816ALJA

But the bottom pivot assembly that comes with this kit is significantly fatter/larger in diameter than the one in my closet door so it won't fit - it seems there's multiple sizes for these things?

Most likely you needThis One, but I would take the bottom jamb bracket out and take it to a local Box store and match theirs to what you have. Sounds like your bottom jamb bracket stretched or got bent and now allowing door pivot to sink.

So it turns out I needed a 3/8" bottom pivot rather than a 7/16" one. I pulled out the old pivot, inserted the new one, put the door back in place, made some adjustments (which were literally painful given how heavy the door was, and the fact that it was practically sitting on my left wrist while my right hand was trying to move things around), and all is good again!

Thanks everyone - I would have been lost without you!