Preventing this at the app store level is hard

>Google, also, per your logic, can run whatever code they like on their servers

True

> Sideload [...] google won't stop you.

Except they do?

https://keepandroidopen.org/

Over the last couple of years they have been installing a massive amount of anti features to make sideloading harder and harder.
We are at the point where I cannot buy a phone and install what I want right away.
There is absolutely no sign that they will stop here.