I think it is possible to overthink this. How real is anyone in our interactions with them? I don't think therapists set out to be less real than anyone else. What they should be doing is keeping their crap out of their clients' therapy. This can be understandably jarring but it's important to allow space for the client to focus on themselves and their process.
I am in the somewhat unusual position of having seen my therapist interviewed on a number of occasions on video. I obviously hear more about him in those videos, so I learn facts about him that I wouldn't otherwise know. Because I'm not paying him to tell me about himself. But I see the same warm, thoughtful person in those videos as I see in front of me every week in therapy. I don't feel he puts on an act or a front. I think it's unlikely that therapists are less real than other professionals you might hire. Its just that keeping their stuff out of the way is an important part of the job.