I think a lot of conservatives hate it becuase they don’t like alternative lifestyles. Libertarians hate it for the reasons you just mentioned. People should be able to do what they want , but fuck off if you think I should be paying for it or punished for disagreeing.
No conservatives largely hated it because conservatives are more religious-based than democrats, and most importantly, because the overwhelming majority of abortions, by a gigantic margin, are simply chosen because "I don't want the baby".
Any poll, data point, etc. you will find will show that the % of abortions that are done by a) rape/incest victims or b) health concerns to the mother or child are very, very small % of total abortions. Combined, the highest poll studies I've shown put them in the 10-12% range of all abortions.
So therefore it doesn't seem super logical to grant "rights" to people, when everybody can agree that literally the only real benefit impacts only 10% of the population of people choosing to do it.
Regardless of what side of the aisle you're on, both parties agree on one central premise... there's too many fucking abortions. You ever met anybody who was happy and joyful to get an abortion? Ever congratulate somebody for getting one? Me neither.
There's too many of them. That's where the focus should be. It shouldn't be on who has the right to do it and who doesn't. It should be rare. It should be for extreme cases that require medical intervention. That's what its designed to be. At no point in time, even when it was a "right", was it designed to be a get-out-of-jail free card to have unprotected sex with anybody and just kill the fetus later. Abortion shouldn't be a contraceptive.
Clearly making it a constitutional "right" didn't do anything to reduce the moral or educational compass of people having them. I don't think taking that right away will change it either. It'll just make it more geographical and more partisan than it already is.
You want the most cynical and "best" argument you'll hear for abortion... there's too damn many stupid people on this planet, and they shouldn't be procreating. There's millions and millions of people on this planet who have a net negative impact on society... their inputs are greater than their outputs, they don't provide economic resources to society, they take in more than they put out, they have low or non-existent moral values, etc. Nobody should encourage this people to have children.
THAT is the argument liberals should lead with. Except its paradoxical, because they pretend to live in a utopian world where everybody is actually equal because everybody breathes air and uses the toilet, despite those being the only things in common people have.