When the kid does get a phone, can you turn off family sharing then put it on their own phone?

Go to Settings > [your name] > Family Sharing > Screen Time. Tap the name of a child in your family group, tap Turn On Screen Time, then follow the onscreen instructions. For information about Screen Time settings, see Set up Screen Time for a family member on iPhone.

Set up parental controls with Family Sharing on iPhone

You may --> Leave Family Sharing - Apple Support

The article discusses leaving Family Sharing for your iCloud family group. That does not necessarily address moving a watch originally setup using “Family Setup” from the parent’s phone to the child’s phone. Disabling family sharing would unpair the watch but it would also do a number of other things likely not intended. There must be a way to move the watch without removing the child from the family group. Unfortunately I cannot find that help page.

Put what?

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