I was experiencing the slowdown in processing that Apple was attributing to an old battery.
Well. Battery took longer to remove than the video would suggest because the adhesive strips broke. But the suggestions for a heat source and dental floss did the trick. When I opened the phone there was a loose bracket that I didn't know where it went. I submitted a question and photo on Ifixit and got an quick response for where the piece had fallen off. All good.
When the instructions say "gently", they mean "gently".