I'm a mental health nurse, and find it's very dependent on personality! Imo, being personable and warm are fundamental qualities for anyone working in psychiatry. Your job is literally to talk to people about the most vulnerable parts of their life. Good relationships with mental health professionals are important. It's not like other areas of medicine imo. Research studies will tell you people have better outcomes in things like therapy if the patient/therapist relationship is good.
A huge part of my job is holding the space for people, and treating people with warmth and compassion. I always focusing on building a good rapport with new patients.
Patients are not my friends, but it's a very human job and people want to deal with someone who feels real. Who would want to disclose their childhood trauma to someone who seems detached? I use appropriate self disclosure, and I take an interest in people's hobbies and lives. I remember the details people tell me. I'll have a joke with people. It's all about getting to know the individual and being person centred.
I find it odd that some people equate being cold as being professional. I actually think it's unprofessional. You should be trying to put people at ease. You can be personable and warm whilst maintaining professional boundaries.
But back to psychiatrists- they see patients less so won't build the same relationships I do with patients, but I'd still expect warmth and an attempt to seem human/personable and not just sound like they're asking a bunch of rehearsed questions. Psychiatrists also work within a medical model, which will make things seem more detached at times.
(There are some cold/awful MH nurses too!! As I said, very personality dependent!! I do wonder why you'd go into the job though if you don't genuinely enjoy talking to others and building rapport!)