I don't personally struggle with eye contact with real people. I don't struggle with it around the therapist either - I just don't have a need to look at the woman and it does not help me to do so. I can look at her if I want to do so or feel a need to do so. I personally only care about the oxytocin idea in relation to animals. I don't look at the therapist unless angry so I am in no danger of being oxytocined by the woman.
My mention of oxytocin is more as a possible explanation why some clients find it so good to look at a therapist. And why some therapists, consciously or unconsciously, seem to employ the stare so much. And I don't think all therapists are trained the same way. Certainly ones I talk to about it in my professional life do not all report the same types of training.

I had a therapist about 15 years ago that I only saw for a 2-3 of months. It was always a struggle with that one but I quit because she decided to sit in silence as I shook - we sat in full silence for two whole appointments and then I quit. (she said I couldn't just quit - she was batshit crazy).
The first one I see now has more trouble with silence than I do at this point. I would rather have silence than her talking at me. If she had stuck to questions I would not mind it so much - but her statements are pointless to me. (the batshit crazy one was not better at talking - I just wanted some direction in the form of questioning).