I had 7 platters total, from two hard drives, and only two of them stuck together in this way. The others somewhat stuck together but it's orders of magnitude weaker.

If I buy an identical HDD off eBay and carefully try to swap the platters, what are my odds of success? I have some experience with such tinkering, and have (destructively) disassembled a few HDD ...

A hard drive works almost exactly like a record player. Data is stored in blocks of 1s and 0s on an aluminum, ceramic or glass platter, which looks a lot like a CD.