Apparently to "keep the peace" and to "protect the children" but I couldn't find any good source on this.

Intuitively it seems to me this is the most counterproductive law ever as living with this doubt is the best way to destroy a family.

Right but I think it's mainly about saving tax payer money from child support by shifting the burden to men.

Protecting the kids I think, because if the dad is not known then the mother will have to pay for the child alone (subsidized by the government). In France around 3% of kids are raised from dads not knowing that they are not the biological father. Personally I think this law is completely unfair but in practice I think the judges will not believe the one opposing the test.

You just can't order a test for someone else (your child) without their consent (so both parents, and a judge because parents don't have absolute rights over their children).

Courts order paternity tests just fine though when there is a reasonable doubt.

The people concerned can always refuse to be tested though.