Those checks don't apply because a.) the children are their natural children, and b.) it is less natural for someone to "love" a child who isn't their own.
When you conceive a child, it's natural for you and the rest of your family to really form a love for that child. That love starts before the birth, which is why you make adjustments to your life during pregnancy, have prenatal checks, buy clothes/other essentials, toys for your baby, etc. You have an innate bond with your child, which is generally a "given". People who are looking to foster or adopt don't have any preexisting relationship, connection or bond with the child and that is mostly why they are subject to checks.
Placing children with people who aren't biological relatives is a *very* serious decision, and one that I think should only happen if parents and other family members lack the willingness to look after the child, or are physically abusive.
If we introduced checks for parents, it would create a system where people are held to unfair standards (e.g, what if the parents are religious and get persecuted for that?) and intervention where it is not necessary.
Personally, I'm very wary of involvement from the state so I would be concerned for people if this was introduced. Beyond the obvious uses we have for government, we shouldn't have that much faith in them...the reality is that they would likely abuse a system like this.
Note: women have midwives in pregnancy, and they can usually tell whether the woman and her partner are up to the task of caring for the child.
