Hmm, I don't know of any in Ohio, though it's not like I know every program in Ohio.
In my residency program, the PD pretty much tossed out any application where the person did not mention any strong motivating factor to be a psychiatrist.
And yes, there is some type of movement among IMGs to go into psychiatry. Why I don't know. Psychiatry is perhaps the one field where fluency is more important than the other fields. There are also several other things that are culture specific. There is a perception that it's easier to get into psychiatry, but in the last few years, this trend is not so true.
As for those two programs that did test me during my interviews, at least back then, I considered them low rate programs. I interviewed at those two places only because of their location. One of them was close to where my then girlfriend, now wife lived. We hoped it could be a place where I could go and not have a long-distance relationship. If you've been on the boards for awhile, that was the same program where an attending there intentionally did a stress interview on me, trying to upset me, and throwing insults. I wrote the program off my list in my mind but only decided to go through the rest of the interview because I already spent the time and money on the trip and I figured it could prep my for my future interviews. That same attending was later accused of assaulting a resident, and all the residents I knew at that program told me this guy had severe anger management problems.
The other was in a similar situation. I have heard from a good friend at that program that I was right not to go there. She described the program as malignant, violating ACGME rules and highly ethnically insensitive. The overwhelming majority of people in that program were from another country and ran that program as if it were in that country---including not following ACGME rules.
But all of this was years ago and things change.