I'd want to see international comparative data before I considered that a statistically significant cause. I'm not saying it isn't, but given the percentage of children born to never-married mothers these days, the change from marital to non-marital births may be your outcome, rather than an absolute decline in births.
It takes a lot of individual decisions to drop a whole population's birthrate. We've reached below replacement in the US despite the fact that, among my friends, six or more children is common and ten is unremarkable. We and all the other religious families (Protestant, Catholic, Jewish, Moslem) are not significantly affecting the nationwide number.
A lot of other countries are not as openly hostile to fathers as they are here. Can’t speak for the situation and reasoning in those countries. And the system doesn’t care about the ones who father children and vanish. Tracking them down takes work and they aren’t about work. They make up for it by treating the ones who want to raise their kids like 3rd class citizens, tracking their every move and draining every cent out of them they have.
Really the drop in birth rate is good news. It means that men in the US are not completely stupid.