I guarantee Columbia will never take a DO. MGH/Mt. Sinai have never taken a DO before. I highly doubt these places have never received applications from qualified DOs. (Except Columbia b/c I don't think a DO even qualifies for an application)
So?
The job of a physician exists for patients. Your average American patient doesn't know the difference between most places. They know football. They know Ohio state is going to a bowl game. They know their local university medical center. They know institutions that have done a good job marketing themselves like Mayo and Cleveland Clinic.
If you ask a person on the street what do you know about Massachusetts General Hospital? They will respond, is that a soap opera? No, wait, that's the hospital on Gray's Anatomy! You have to say the name Harvard. And in doing so does your average person know if there is a difference between Harvard, Stanford, Yale, Mayo, Cleveland Clinic, Wash U, Vanderbilt, Emory, etc for medical training? All are private institutions.
Massachusetts is not a place a physician should want to practice in. This article exemplifies that.
http://blogs.wsj.com/health/2009/11/27/will-a-move-away-from-fee-for-service-control-health-costs/ MA is now shifting their state towards a glorified capitation model. Capitation failed, and failed hard during the 90's. It most certainly won't succeed now. This isn't the least of the issues MA is facing. A keen physician would know to stay away from that state, and many have left. Recognizing this, why would a graduate want to stay on as faculty? A sign of a good program is one that has a decent retention of their own graduates. I suspect this will greatly be changing in that state as graduates seek out more fruitful states.
Mt. Sinai and Columbia are both in New York. They are astronomical for cost of living. I don't know if you have been paying attention to some of threads on the forums about student loan debts, cost of living, the economy, and psychiatry salary. Our economy will not recover for a very long time if at all. This is going to be the new norm. Declining world resources, rising world population, and possibly even changing environment all have economic impacts that no country is going to bounce back from readily.
Psychiatry isn't going to be increasing in income. Do you really want to live in a city you can't afford during residency and very likely post residency? Insurance companies or the government will not be clamoring to increase reimbursements for physicians. Americans don't care about a physician being paid appropriately. They seldom know the difference between PA, (D)NP, or Physician.
If you think you will opt of the system and do cash only catering to the people who know the value of paying for a physician do you think they will know the more prestigious psychiary programs? U. Cinncinnati is quite established with research, but would a patient known the difference compared to OHSU? Would a patient know if they are buying the best psychoanalyst if they trained at programs like Iowa or Wash U that are more biologically orientated? If this is a goal, it is best to know what the people in your practice area recognize as "the best". But at the end of the day, your personal reputation and word of mouth will matter more.
Getting back to your points about DO's at those institutions, I say it really doesn't matter and Osteopathic Physicians aren't missing out on much that can't be obtained elsewhere, too. It just means they are going to programs that are much better overall in the numerous other variables that mean a lot more in the real world.
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