I really only know what it's like where I trained, and where I've worked since. I've never been able to get someone into a psychiatrist in any reasonable time frame. If there are places where this is not true, then I'd love to know where these places are!
I do treatment resistant depression, bad anxiety, PTSD, OCD, and bipolar all day every day. Some stable schizophrenia/schizophreniform, not much though. ADHD from start to finish. I have a hard time with personality disorders (who doesn't). Of course, plenty of addiction (one of the most common chronic diseases ever).
There's a lot of overlap and psych symptoms with dementia, TBI, MS, Parkinson's, somatoform, centralized pain, fibromyalgia, etc. etc. and other neuro or pain disorders for sure.
I do all outpatient. I would be completely lost with difficult inpatient or ER psych.
Edit: I really think a family or IM trained physician can do the vast majority of outpatient psych. The stuff I don't do in psych is really hard, and I wouldn't want to do. I wouldn't get a lot out of a psych residency because I can already do what I want in psych currently.