I don’t think it is so clear cut either way. but does that really matter? A law, even if passed by a slim majority, will still officially represent America. I think the partial abortion ban that was passed sends an important message.
That was a symbolic gesture, basically "throwing a bone" to the base. PBAs are a rarely-performed procedure, the vast majority of which happen when the pregnant woman experiences traumatic medical complications and her life is in danger. The myth of mobs of 8-month pregnant women rushing to Planned Parenthood because they were depressed over their inability to fit in their prom dresses was exactly that, a myth.
Most Americans will support some limitations on abortion, I'm not denying that. But if the question is simply, "Does a woman have the right to end an unwanted pregnancy?" the consensus in this country is yes.