Short answer: Yes. I've worked in psych in Virginia and Pennsylvania for quite a few years.
This is what happens in 2 different types of situations.
A. Person is abled bodied and medical transport isn't deemed necessary. The correctional officers/ police would transport people with criminal charges, convictions, etc. to and from psych facilities all the time. Usually the person is handcuffed. The procedure itself is interesting. The CO/police arrive at a psych facility for pick up or drop off. They have to surrender their firearm at the facility and it is locked in a secure location. They then go with psych facility security and staff to fetch or drop off the person. In a private location shackles are applied or removed. Staff goes back to where the officer's firearm is secured and it is returned to them. Now does every facility follow this protocol? Probably not. Do many follow it for staff and patient safety: Yes.
B. A person is very sedated. This would be when medical transport would be called just on the off chance the person loses their airway during transport. But like I said before, the person would be strapped in 4 point restraints to the frame of the stretcher. They aren't going anywhere even if they "tard out" in the event the Haldol and benzos wear off. I've never seen COs ride along. I'm not saying it never happens but, I don't think Chris is such a special snowflake to warrant it.
Also you made a comment about the police probably have time dealing only with speeders and DUIs in rural areas for this type of trip. Actually the opposite is true. It's a rural area. Less people equals less police and less taxpayer dollars. I've lived in rural areas in both states that don't even have full time police and either have only a few part-timers or rely on the state police. Trust me, they have much more important things to do.