Hello! I'm at Columbia currently and originally applied because of the dual degree program. I have since decided not to follow through and just get my MSOT. Columbia was the first and until last year the only program to offer a dual degree in MSOT and MPH so the Wash U program must be new.
The way it works here is you finish your OT degree and fieldwork first then do a year of coursework in public health and then do a masters project in public health. So it isn't really an expedited program, it is two full programs but they shove more courses into a year than usual. The public health program is good hear from what I hear but nothing earth shattering. The issue is that by time you finish your MPH you have taken a full year off of thinking about and working as an OT which can make it difficult to get a job.
I decided not to do the dual degree because there are few options where you would really use both of them. It is a lot of money (around 20k a semester for tuition) for an extra degree that won't necessarily increase your pay, especially if you are just starting out. I have a couple friends interested in getting a public health degree after they graduate but they also don't plan to work as a traditional OT. They want to do more broad global work where they design rehab programs in developing countries or want to work for places like the UN where they are more focused on policy. So, if you want to work at a traditional OT site, I would personally recommend not getting the public health degree. But, if you know how you would use it, then go for it.
The best thing you can do is contact the schools and ask what the benefits are of getting a dual degree are and discuss what your future plans are and see if an MPH is right for you. They were very open with me here.