The laws state that a child born with in the marriage is the child of both parents, DNA results not withstanding. This is a law passed years before there would be ways to definitively track genetic heritage.
The laws may need to be changed, but this is what they are now.
As it sits the child is his because legally the child is his, not genetically.
Since genetics is ignored, what about the financial responsibility for the children following a divorced woman in a new marriage she may engage in.?
New husband takes over the responsibility.
Proof of paternity is not the issue and DNA is just better proof. The law was not written because proof may be lacking or questionable. The law was written that way to contravene the proof.
If you had no hard proof, like DNA, to prove paternity then you cannot disprove it either thus no need for a law. But if you can prove or disprove paternity then the law makes sense in that it does what it supposed to do which is keeps the child from bastardy and effects.
There is no penalty for her trampish behavior. . .no disincentive at all.