Assistive Touch home button on my mom’s iPhone SE comes on randomly, won’t go away no matter what!

My mom’s iPhone SE (2000) will not let us turn off the on-screen assistive home button permanently. It keeps popping up, after a complete wipe and re-install. We turn it off in the Assistive Touch panel, and turn off the assistive shortcuts, and it keeps popping back up a minute later. Just installed the latest upgrade before the wipe and re-install. No luck.

I know some answers on here for the exact same issue have been “it’s the battery,” but that makes no sense. There’s never been a decent solution to this problem, as far as I can tell.

Does anyone have any better suggestions than “your stuff is old, go spend $500 on something new”?