Wow, that version revert feature is really cool! Unfortunately, it didn't solve my problem. But! I did figure it out. I'll share in case it's ever useful to someone else:
The old JSONs that weren't working contained mesh animations. When compared with other (equally old) mesh JSONs that were working, we noticed there was a small difference between them. The broken JSONs all contained the term "skinnedmesh", while the functional ones simply had "mesh". I removed "skinned" from the broken ones and magically they work now, hooray!
Thanks a lot for your guys' help though, and the quick response!