Unfortunately as a practical and legal matter that is false. First, physician incentives are aligned to deny care: they have a defense for denying care ("my lawyer isn't clear that I have authority to do this") and the woman has no recourse. Second, there is a simple matter of skill and availability. Fewer facilities allow abortion; fewer OB/GYNs are skilled at doing it safely. In my pregnancy I wanted a perfectly reasonable and legal thing supported by medical evidence and was unable to find a doctor in the state to provide it (vaginal breech birth as opposed to forced C-section).

When you are pregnant, and particularly if you are experiencing complications, you do not have time to shop around and convince people and schedule in advance and all that. You are constrained by the spatiotemporal availability of a skilled medical professional.