It's hard to say. I would actually take the opposite side and say get a MPH/MHA. If you know that's what you really want to do, I think the programs are much more tailored to what administrative fellowships are looking for and I think at some places this degree is probably still more respected than the MBA. I think it's true that many (definitely not all) will take an MBA as a substitute, however, the recruiting process for these things is a whole different ball game. They almost always require letters from program administrators, faculty, etc which isn't the same way that MBA programs tend to work. So if you do get an MBA, getting it from somewhere that's familiar with the hospital admin fellow recruiting path I think would be helpful.
The other thing to consider is placement, because I suspect these programs actually have better placement than MBA programs. I know University of Minnesota (probably also Michigan), for example, has 100% placement, which is something to consider. Also, I'd be willing to bet that you actually find the same amount of crossover career-wise. What I mean is that in MBA programs, you will find 1-2 people going into hospital administration and in MHA programs, you will likewise have 1-2 people going into industry (including in some of the fields, like pharma, mentioned above). They will not be business roles, but from what you've said, that's not what you want. It seems that most people wanting to be involved on the delivery side don't really want to go into pharma/devices anyway.
Geography might be something to consider as well. As you get down the ranking list, some of the opportunities may not be at big hospitals, which might mean you're managing a unit in a small town somewhere. The kind of opportunities that tend to show up at MBA programs are from bigger systems, which often are going to be in bigger cities.
Tuition is a wash if you go to a top MHA (same tuition as top MBA), which might tilt more favorably to the MBA side.
"Egotism is the anesthetic that dulls the pain of stupidity." - Frank Leahy
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