Math Overflow is not part of any "model". It is not part of the Stack Exchange network. It is not run by the Stack Exchange team. It is not subject to any of the rules and processes of Area 51. It is a private site, run by a private group of people, that happens to use an older version of the Stack Exchange software from back in the day where Jeff & Co allowed it to be licensed in that way.
I respect the people over at Math Overflow for having been on the bleeding edge and creating a community that is successful in its own right. But there was never a proposal. The definition was never vetted. The operation of the site is entirely up to the whims and fancies of its creators and moderators. And between MO and Math.SE (which shares certain members), there's been a whole lot of drama, and they've managed to rub several people the wrong way. That's OK, that's their prerogative - it's their site.
Math.SE is simply a different animal. It is not the beginner version of Math Overflow, as some of you so dearly wish it to be. It is all-inclusive, like Stack Overflow. It says right in the tagline that it is for all levels of math, and yes, they do suggest that if you have a research-level question, you might get a better answer on Math Overflow because that's where the professors and grad students hang out, but no legitimate math question is off-topic on Math.SE, and MO can't migrate questions there either.
Math.SE is also the only math site that is actually part of the Stack Exchange network and thus subject to SE policies. Math Overflow, for all practical purposes, simply doesn't matter as far as Area 51 policies and decisions are concerned. They aren't some special exception or experiment that people tried. They're just a group of very smart people who formed their own elite community using the original SE 1.0 software. If SE 1.0 had never been released, they probably would have eventually turned to one of the Stack Overflow clones, or maybe even a forum, who knows. The point is, that community would have existed and arguably already existed independently of Stack Exchange.
That's not germane to the issue of whether the Stack Exchange network should include two sites for every branch of science with communities who simply wish to exclude others. So just stop talking about Math Overflow already. Please.
The problems are really rather clear-cut:
The line between "research-level" and "beginner-level" questions is blurry and subjective. It's easy to point to extremes, but in the middle it can (and usually does) become a pissing match with both sides walking away frustrated and angry.
Even if the boundaries are well-defined, there will be a non-trivial subset of people who simply don't or can't understand them. There already tends to be confusion between, say, Super User and Server Fault, and their scopes differ by more than just skill level.
The experts on both hypothetical sites are the same group of people. The difference is, essentially, that fewer of the experts will participate on the beginner site, and those who do will participate less than they do on the research site. Low/tentative commitments are a serious threat to the survival of a site proposal.
Segregating by skill level runs completely counter to the Stack Overflow / Stack Exchange philosophy. It's been enshrined in culture and policy since the very first elevator pitch. The SE model is: Most traffic comes from search engines (not regular members), and no question is too basic (unless it's trivially answerable with a well-known reference source like Wikipedia). You may not like or agree with this approach, but again: If you want to be part of SE, you need to take the fundamental SE philosophies and practices to heart.
There's simply no evidence to date to indicate that skill-level segregation is actually beneficial to either community. I'm appealing to facts and data here, not opinions. Area 51 has already staged several sites and the data is pretty clear: larger communities do better as long as there's a clearly-definable group of experts and significant skill overlap (i.e. same profession, same university department, etc.).
In closing: You want to put the onus on us to prove that the segregation is bad, but that's not how it works around here. There are plenty of problems created by insulated information silos, but in the end, that's all irrelevant; getting your own site isn't a civil liberty, it's a privilege granted to those who can prove their worth.
You tell us how your proposed division is going to benefit either community. Prove that it's going to be more than just a dump site for questions deemed unworthy by the research community. Because we already tried that a few times, and the results were not spectacular.
Around here, the default mode for similar proposals is "merge". The simplest and most important reason why skill segregation is bad is that no evidence has been presented to the contrary. As I've stated in previous comments, if you want our support, then you'll want to demonstrate (a) that they are truly separate and distinct communities with no real overlap, and (b) that a "beginner-level" proposal is actually viable on its own merits.