I've got a bunch:
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It should be possible to manage my kids' access to all services from third party apps. Like, I want to say "no voice comms anywhere" without having to fuss with each terrible janky UI.
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There should be required messaging standards. Let me choose emails (or texts, or whatever) for friend requests, weekly summaries, etc.
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Tell me who manages my kids' friends accounts.
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There should be a standard, easy easy to register an account for parental controls. I don't want to dig through out-of-date support docs to try and figure out how to link my account to my kids' accounts.