I think you are right in general. But I also think it is a difference if you drain the battery controled for some seconds/minutes to 0% and directly start to charge, or if you let your battery die for some weeks by self-discharge…but it’s just a guess.
I think we can conclude, that for all reported problems here, “only” the phone is hot, not the battery itself.
If it is located under the written text “yours to open, Yours to keep” of the FP2 back, it is for sure “only” the CPU unit.
Then you should look in the software for a bad app, which keeps the CPU busy, whats results in hot CPU and fast draining of the battery, thus short lifetime (as expected under “full load”!).
Gave you a try to disassmble and reassembe all easy parts to teardown and check? Maybe just a loosen contact somewhere?