You cannot argue that eventually a fetus becomes a person.

No real argument there. But "personhood" can be a troublesome notion to define. I've found it easier to frame it as "At what point should the government confer certain rights to a fetus/person?" After all, there are many rights that are age gated and no one really balks at that (driving, smoking, voting).

I do think abortion should unquestionably be allowed up to the third tri mester. It's just after that where it starts to get into that gray area for me personally.

This gets back to doctors being overruled by the government. If a doctor recommends terminating a pregnancy the day before they're due date, who are you or the government to contravene that? The licensing isn't the government overruling their decisions. That's the government's check to ensure its citizens won't fall victim to some quack rubbing then down with crystals and essential oils while calling themselves doctors.

This is the point opponents like to use the "abortion as birth control" and try to argue these late term abortions should be banned to prevent that. That's just simply not the case for late term abortions though. Women are not carrying a fetus for 9 months only to up and change their mind right before it becomes unequivocally murder. Again, the hypothetical is this is a doctor's recommendation. A doctor would not be recommending terminating that late simply because the person felt like it. Things would need to be catastrophically dire for the mother or fetus for that to be recommended that late. On top of that, a tiny fraction of abortions happen in the third trimester. The implications for malpractice are huge if they recommend dangerous treatments that are not outweighed by the benefits. You have espoused at least a modicum of faith in the government to identify "the greater good". Would you extend that same faith to doctors? Assume they will mostly act ethically and professionally and only recommend abortion if it is truly the best option?