You have noted the inconsistencies with the claim that CPS needs any paperwork to seize children they can't just make up on the spot.
The courts are tools the wealthy can use against the poor, and institutions like CPS are possessed of vast wealth. Family courts are the most corrupt of all in the system.
If your family is threatened, or has been attacked by the CPS, get the best lawyer - and get references from people you can trust! - to fight.
Don't play paper games with your kids lives. Don't be a paper tiger - be a real tiger mom!
Your kids are worth far more than money.
Just ask them.