Yes things like this exist now, f-droid is an example, it allows users to download android apks without a google account.

Other options include downloading the APK straight from GitHub..

The play store and apple iOS store, their respective companies decide what is allowed and not allowed.

Well the only thing centralized about the app stores is that the companies are sole deciders if the app should be approved.

@jb55 went through that with Damus and iOS over the last year or so.

So the feds can pressure Google to remove an app, either with asking or court order like samourais case.

Tldr.

Yes with third party app stores, like f-droid, or downloading the APK from the source,GitHub, you bypass android appstore stuff.

iOS doesn't currently allow you to 'sideload' apps, 'sideload' just means installing from somewhere other than the official play store.