This kind of policy is why I refuse to buy apple. My 14yr old daughter has an iPhone and I cannot administer it as a parent without owning an iPhone or similar device myself. A basic Web UI is all that's needed to access the screen time settings, but nope. It's clearly by design, you're either all in with them, or they dont want you.
If you could administer it from the phone, it would be possible for an enterprising child to remove those same restrictions on that device (how many stories have we heard of kids removing the parental control software from a computer?)
If you could administer it from the phone, it could be used to partially lock you out of your own device if done by a 3rd party (loan to a friend, and you get it back and you can't use any of the other things on it).
If you could administer it from the web, someone else on the web could use it to lock your out of your own device.
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Having one device owned by the parent managing another device owned by the child ensures that only the parent is managing the child's device and furthermore that only a set of devices that have been set up under the same Apple ID family can be administered.
If you can administer it from a webUI that doesn't have authentication back to another device you hold with the keys for managing that Apple account, then so can an attacker.
For another phone, yes - that would work... but it needs to be one that holds the private keys for the apple account.
Those keys are held by a part of the apple hardware that prevents them from accidentally leaking outside. Because of how "find my device" works, leaking those keys would allow a 3rd party to track you (or access your stored secrets associated with your account) - and so Apple has been very careful with the hardware and software securing those keys.
From Apple's perspective, the ability to administer the family plan from a web UI is inherently risky and possibly privacy violating - neither are things they want to let go of. Part of the brand value is that it isn't risky to use their devices and that they make the privacy of the people them something that those people trust.
And so, consider, that you're asking Apple to allow someone to log into a website and use a password (possibly compromised) to restrict the functionality that a given device has without being able to verify back (send apple verify codes to devices held by that same account) that the person making the changes is one who should be able to do it.
Apple allow me to buy stuff with my apple account on other devices, apple music, iTunes, etc. So clearly if I'm giving them money they are just fine with non apple devices. All I want to do here is change thr screen time allowance on my kids phone. Its totally possible, and security is clearly not the reason if they allow 3rd party devices for financial transactions.
