Can an Apple Watch be paired with the child’s iPhone?

The Watch can be paired to the child's iPhone. Once paired it cannot be paired to another device unless it's unpaired from the child's iPhone.

You can use Screen Time over Family sharing to impose restrictions on it, it does not need to be paired to your own iPhone.

click here ➜ Set up Apple Watch For Your Kids - Apple Support

It sounds like what you want is Family Set up.

Set up Apple Watch for a family member - Apple Support

The watch can be paired to one phone only.

You don’t want to do that…

What is it you want to control about it, exactly?