You don't babysit your own child; it's called parenting.

That would not be payment for child care.

She's already paying taxes on the money she earns, so why would they want to turn this into taxable income for him to report, eating away into the funds? The more they run the money in circles, the less they end up with.

Look at Figure A:

https://www.irs.gov/publications/p503#en_US_2021_publink1000203267

"Were your payments made to your spouse or the parent of your qualifying person who is your qualifying child and under age 13?"