What do you think of the idea for a deadline for a man being able to renounce paternal rights/obligations be tied to the deadline for when a woman can decide to have an abortion?

I recently saw a post on r/ask basically asking if a man should be financially responsible for a child even if he never wanted one. This got me thinking, and I came up with the following:

I think the optimal outcome in that scenario would be that the man should be able to sign the rights/obligations of the child away if the fetal development is still within the time that makes it legal for a woman to choose to abort said fetus.

For example, in a state where abortion is legal up to the second trimester, paternal rights may be signed away up to the second trimester. If said state had abortion rights up until birth, then the paternal rights should be able to be terminated up until birth. Maybe have an exception regarding a father charged with rape of a mother, in which the proven charge of rape waives his rights to terminate paternal obligations.

Once a father terminated paternal rights/obligations, he may not have his paternal rights/obligations reinstated without going through the normal adoption process.

This may also have the side effect of encouraging the loosening of abortion laws in my country (USA), as now it gives both sets of non-birthing parents a personal reason to care about the issue.

Are there any reasons that I am not thinking of why this would be a bad idea? Am I making any assumptions that I should not be making?