>but it seems to me like primes are more common than zooms for portraits.
1. Your lens needs to be a certain distance to the subject, or else the perspective will cause undesirable distortions.
2. People just sort of found out the 85mm focal length forces you to be at a good distance to the subject, and therefore good at portraits.
>Q. Why primes though?
People would rather hold a smaller 85 prime than the 70-200 zoom. I think it's that simple. Also F1,8 and F1,4.
>Does it just come down to the sharpness
You could actually just use a 24-70 zoom, and use the 70mm end, and stand at a similar distance as you would with an 85mm.
This will give you the same perspective, and avoid those big noses.
But your 70mm would have more environment in the frame, compared to just having the subject in the frame.
Thus, in the 70mm you get less resolution of the subject, because the subject occupies less area in your censor than otherwise with an 85mm.