But let's not mince words here, we are ultimately coercing men to do something against their will. Like paying taxes. Like abiding the rules of law. This isn't something without precedent...

If being a parent, or being responsible for a child which came into being as a result of your actions, were as compulsory as taxes and the rule of law, abortion wouldn't even be a thing, where health isn't a concern. Yes, I understand, "her body, her choice," and I agree with that, because I don't see any way to reconcile bodily autonomy with parental responsibility. (Ideally for me, both parents would accept full responsibility for the child where the child is a result of consensual activity. Unfortunately, that's a pipe dream.) Abortion has most definitely been used to opt out of becoming a parent because the potential mother decides she's not ready. Extending that right to the father, giving him some say in whether he is responsible for the kid, is more or less laying bare the reality of what abortion is used for, when health isn't the primary concern.

I'm saying that when not every woman has the luxury of being able to terminate their pregnancy due to access issues, I'm not sure why equality means that every man should be able to terminate their relationship to a child with the flick of a pen.

Perhaps part of the surrender could be paying the full cost, or at least a portion of the cost, of the abortion? Maybe the surrender could be contingent upon securing an abortion for the mother, if she chooses to abort. That way is isn't as simple as every man being "able to terminate their relationship to the child with the flick of a pen." There's definitely more discussion to be had around this, but it in practice it probably wouldn't (and definitely shouldn't) be as simple as you portray it in your last sentence.

I see where you're coming from, but I think that where there's a place for non-health-concern abortion, there's a place for LPS. We just have to find a nice balance where it works out evenly.