Then, when I got to see the gynecologist, who was the best I've seen yet, I started my gender critical spiel.

I began with the new patient form I'd filled out, asking her whether she really considered biological sex changeable. She said the office had put the option of claiming transgender on the form to be considerate of transgenders. I attempted to explain why normalizing the idea that men could identify into womanhood was so dangerous for women.

Afterwards, I specifically talked about the dangers of girls being sucked into the transgender craze, thinking that she, as a gynecologist, has probably noticed.

She was obviously very intelligent, and listened for a bit, but the minute I mentioned that the Biden administration was behind the changes in law, she shut off, snapped that the office was "free of politics" and then continued with the examination as if I hadn't said anything at all.

I'm going to be honest with you, it's pretty reasonable for her to shut you down there. Her talking about politics can get her more in trouble as it's not related to medicine and it's purely political. She can explain why there're stupid changes to forms, and listen to you talk about the gender delusions, but the longer you talk, the more she has to listen and try to refrain from stating her opinion which can affect how she does her job (or if she can keep her job or not if it gets out that she has an unpopular opinion about any party). She may not even be in any position to afford a potential lost or attack by stating her opinion unlike other people (e.g. Toronto University's professor Jordan Peterson).

By the looks of your post, your conversation with her seems largely one-sided with you talking and her just listening. She probably tried to refrain from stating her thoughts as that could put her in a "bias" frame and can cause some issues with the job. This issue is not unique to the medical industry, when you're working in customer service, you can't openly state your political, religious thoughts lest you are fine with losing your job.

From what I understand medical pratcitioners and medical companies are trying to take the path of least resistance because they want to get their job done, they're after money. It doesn't really indicate if your gynecologist is a GenderCritical, a TRA or support Biden.

Although if it's their personal private practice that they own and not part of a larger company, then I would suspect they are TRA while trying to look politically neutral (though they could easily have GenderCritical views, but care more about money than trying to actively counter the TRA delusion). If they're part of a larger company and are simply employed there by someone else, then you don't really know. Go for a practice which is privately owned and run by a medical practitioner who doesn't have those stupid things on their forms if you can, less likely those that don't follow popular social trends are more likely to not care or support TRA stuff. You don't actually know their beliefs unless they state their beliefs to you outside of the professional environment.