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Apple made a guide for how to use Mastodon on iOS:

https://apps.apple.com/us/story/id1668483560

So conceptually they are fine with it. And there is nothing you stopping your running servers on your devices.

Servers that can’t run continuously aren’t very useful, though?

Whether you think they're useful is one thing, but the point was that they're not in any formal sense forbidden by Google or Apple.

Something like https://www.manyver.se/ ? It is available on Google Play, Apple App Store and F-Droid.

It seems they have had some issues with Google Play and the App Store with their weird catch-all policies, requiring integration with Google/Apple login etc. but they have managed to work around it for the time being.

How is the relevant to the conversation? Sure, we have very powerful processors in many devices in our home that are idle much of the time. What, exactly, is your point.