No doubt. But that isn't the question. That's the beauty of questions on polls -- they can be phrased in ways that give an answer your looking for. I'll bet that very question would get a solid majority in my state. Yet my state put abortion-on-demand into its Constitution by a 72-28 vote.
The people will absolutely accept "curbs" on abortion. No politician anywhere in this nation (save perhaps San Francisco, Berkeley, Seattle or my town, Olympia) would be harmed at all by passing a ban on partial-birth abortions. In most places (again not the west coast), politicians wouldn't even be hurt by a ban on 3rd trimester abortions.
But a ban on all abortions except for the life of the mother, or other equally dire circumstances, would create mayhem and havoc in the streets of urban centers where abortion is most popular. After the experience with Prohibition, politicians are going to walk very carefully when taking social "rights" away from the public at large (now I fully understand that you or I would not consider this a "right" -- and in a vacuum we'd be absolutely correct -- but there is a sizable part of the population that does consider it a "right" and that's what matters in this case).
No politician is going to take that step while emotions run that high on any issue. Frankly, the politicians are hoping to punt this issue long enough so that technology solves their problem for them...after all it was technology, medical technology, that had a big hand in creating this problem in the first place.