There's a Family Sharing website
Family Sharing - Apple
Eg: says
" As the organizer, you’re responsible for adding to and managing your Family Sharing group. (If you turn on Purchase Sharing, you’ll be billed for existing and future media purchases and subscriptions made by all family members.) And if you set up Ask to Buy, you’re responsible for approving or denying purchase requests from members who aren’t yet adults."
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Also says: “Family Sharing requires a personal Apple Account signed in to iCloud.”
(1) If you have a Library membership card from a Library that has computers for use by members, you may be able to access and manage your Apple Account there. I don't know how you would receive a verification code from Apple though, so you can prove you are you. Perhaps you have an android phone with a number that Apple trusts ?
(2) Also: If you had iCloud access on the old phone:
You may or may not be able to sign in to iCloud.com https://www.icloud.com/ with your Apple account details - likely the same details you use to log in here to the community.
(3) Yet there is no way I know of to change family sharing settings once they've been set up, without access to a parental device or child Apple device that allows access to the settings..
(4) If the Ask to buy requests were coming via email, you'd likely be able to log-in to your email account and answer the emails at a Library too.
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Re: " ... I have to remove kid safety restrictions just so they can use the devices."
If the children have their own Apple Accounts, as Apple suggests, safety restrictions are likely to still be turned on by default for child accounts.
