This question is partially a red herring. I say “partially” because certainly some people who are pro-choice do indeed consider it important to their view on the subject that the fetus ( / embryo / blastocyte) is “not a person”.
But only some of us.
Me, I’m comfortable making the following stipulations:
a) To be human is to be a person. It’s a social construct, a category WE’VE invented that only has whatever meaning WE glue onto it, yadda yada, but sure, why split hairs (even potential hairs)? A fetus (yea, even an embryo or blastocyte) is a person in a sense that a fingernail clipping, an entire finger, a spermatozoon, or a dead body are NOT a person.
b) Abortion is killing. It is other things, too (medical procedure; termination of unwanted medical condition; exercise of the right to make one’s own reproductive decisions; etc) but except in the case where the implanted conception-product is already dead, abortion kills it.
c) Abortion is therefore the killing of a person.
I’m pro-choice. I think abortion should be legally available to any pregnant person. Period. No qualifiers. I’m OK with Roe v Wade as a compromise.
I ask for the following stipulations from anyone who is NOT pro-choice:
a) Not all killing is murder, or even immoral. While I am not going to point to any specific example as being just like abortion (nothing precisely is), it is generally held that to kill in one’s self-defense is neither immoral nor murder; it is held by the majority of people, if not by everyone, that to kill in combat as a soldier in wartime is also neither immoral nor murder; and in triage considerations (military or medical) it is generally accepted that it is sometimes necessary to cause the death of some via the taking of action designed to save the lives and/or establish the goals of others.
b) We have a general abhorrence towards the killing of human beings, but it does not have the status of an “absolute value”. When and where it is wrong, it is wrong for a reason. Normatively it is an assault upon the person who is being killed insofar as that person has a consciousness containing plans, history, and intentions, all of which investments are lost prematurely when the person is killed. Normatively it is also an assault upon the person’s family friends associates and community as well, insofar as that person has understandings and skills, relationships utilitarian and emotional, and has likely been the subject of investments of time and resources, all of which are again lost prematurely when the person is killed. Perhaps there are other reasons whereby it might be concluded that it is wrong to kill a person, but the argument has to be asserted and defended; establishing simply that a person would be killed does not ipso facto make something morally wrong.
c) There may therefore be legitimate grounds and legitimate perspectives from which any given abortion may be deemed appropriate and necessary and not immoral. A decision is therefore required, either across-the-board (applying to all abortions) or categorical (in which case abortion situations must be categorized, and therefore categorized by someone), or individually (in which case each individual abortion situation must be evaluated on its own merits independently). Therefore it is a matter of critical importance to decide who should be making those decisions, and whether to try to make them once for all situations, to create a categorical structure and a decision-making structure for evaluations situations to categorize them, or to deal with each individual case on its own merits.