IMHO forensic psychiatry is like being an actor with a college degree but getting offered to work in something stable, paying well that is not acting. Eventually such a person stops acting other than as a hobby.
Forensic psychiatrists, if doing sizable amounts in the field, can't simply do inpatient unless the department they are in will allow them to make depositions and miss inpatient hours, often-times forcing another doctor to cover for them. If you are doing private practice, you have to free time in your schedule to do forensic work and if you have very sick patients this becomes difficult. You might be offered a very serious case that could have you working over 30 hours a week on that specific case despite your existing workload, then you might not have any work for weeks.
I did a homicide case while working 55 hours a week and having a newborn kid. My wife was ticked off because it made her a defacto single parent during that time. The case (thank God) was one where there really was no defense for the client that hired the lawyer that hired me. So it boiled down to me reading the records and telling the lawyer why I thought there wasn't a psychiatric defense. By then (and I forgot the exact amount) it came down to about 20 hours of work. Had there been a defense this could've been over 100 hrs of more work because I would've had to write a detailed report based on thousands of pages of records, driven to the opposite end of the state to testify, and prep for a cross-examination.
Doing something like this when you're already working more than a full-time job is stressful. I think most psychiatrists will simply get a full-time job, and with life taking over, just forget about forensics as a paying profession.
It is also possible to get a forensic job that is steady but those types usually end up paying just as much as a regular clinic job. E.g. in Cincinnati there is a practice called Court Clinic and the way Cincinnati's laws are, several defendants get a mental health/competency evaluation. (Most localities have little requirements for such but Cincinnati has lots of requirements). It's to the degree where psychologists and psychiatrists there can expect people to evaluate as surely as a city ER will get patients. Just that you don't make more money there either, nor likely exercise things that likely drew people into the field. (Those evals are similar to just doing an H&P, then typing a report in layman's terms).
I worked on a forensic unit for a few years and while I used my training there, again, it made no more money than had I just been a non-forensic psychiatrist. While I was held in higher regard, the politics of that situation prevented me from more money. We had a chief clinical officer that didn't want anyone making more money than others so he could avoid getting an angry phone call from a lower paid doctor.
That's where the connections help. If you have none, you have to work out of your way to get them. It can be done, but we're talking months if not years, and for most it's just easier to work your regular job.