Well I guess my awkward experience was getting one in the first place.
I just recently moved to a new city and was needing to get established with a primary care physician. I got a few names, and just randomly picked one close to my house. There were no red flags when I called to make an appt - the receptionist simply asked why I'm coming in, set the appt date, and that's it.
When I get there, nothing really seems unusual - the typical adverts for birth control, lung cancer, heart disease, etc. A few other female patients...no biggie. They call my name, take my weight, my vitals, put me in a room, give me a gown, ask me why I'm there (to establish a new patient-primary care relationship). Nobody says anything. Finally, the physician walks in, (again) asks why I'm there, and we have friendly, non-medical history give-and-take for a few minutes (i.e., how I'm liking the city, work, etc). After awhile, she finally let's me know that she's actually an OBGYN (obviously geared towards women's health/women's primary care), and not a 'men's health' primary care physician. However, after saying that, she says that since I've just moved to the area, and have no serious medical issues, she'd be OK to refill my meds in the interim until I get established with a different primary care physician. It's at that point that my MTF status comes out. As she's reviewing my meds, she notices I listed E2 and inquires about it - I tell her about my HRT and transition, my endocrinologist, etc. To make a long story short, she says she would be OK seeing me as a patient, and would be comfortable co-managing my HRT, if I wanted her to.
When I think about this experience, it's kinda funny, kinda fortuitous; my friend thinks it hilarious and a testament to how oblivious I am to some things...evidently, there are online reviews of her...all by women (pregnant women, post-menopausal women, women needing birth control, women with very specific women's health issues, etc...but all women).
I'm still early in my transition with HRT, so I'm not too sure about going to her exclusively. On the plus side, she's only ~20 min away from me (as opposed to my endo, who's ~1 hour away), is really nice, is willing to take me on as a primary care patient, and is comfortable with HRT and my transition. We'll see.