There is no real way to update the game without installing the game again.
One thing I'd say you're doing wrong is uninstalling before installing again.
That makes you lose your saves, you're supposed to install the new APK as an update.
The update might fail, often because you don't have enough space.
But check the appstore and get Split APK Installer to install the game, if it fails at least it will actually tell you why.
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Yeah, I know.
But for most people just updating the game works.
Only time it didn't work was when Arc lost his dev keys and the thing was complaining the signature wasn't the same.
With this game normally it works and the only errors while installing tend to be not enough space.
Some seem to be fixed by installing with SAI for some reason, I recommend it because at least it will tell you a proper error message instead of just saying "No, can't install"
But yeah, ideally people would backup their saves.
It's just a bit more involved, and not every file manager works (at least back then when android 11 and the restriction were new)