If you don't have a clear mechanism by which a startup could solve this problem, I suspect the result would be the same structures that already exist, but with better marketing and a Web 2.0 interface. If anything, this would make the problem worse, because it creates a company that now has a vested interest in keeping the status quo because it's part of the status quo, and that company will be manipulating politics and public opinion with its marketing.
A second problem with this sort of company is that it often doesn't really understand the industry it's "disrupting", so they end up not solving a lot of the problems which older solutions solved.
At a more fundamental level, I think that a company is always going to be motivated to make money rather than solve a problem. Even if intentions start off well, the people in the company will believe that the company is good and needed, so they'll make choices that give up a little of the original purpose of the company to make the company survive, because if the company doesn't survive then it can't do any good, right? And after a bunch of one-degree deviations from its original direction, the company is going a completely different direction than its original purpose and isn't even geared toward solving the problem any more.
Sure, "We're going to save the world" is great marketing, but at this point I simply don't ever believe that a startup is a solution to a major problem like this.
Having an open, accessible repository of scientific data sounds like a good thing, but I doubt it would help the problem of reproducibility.
The biggest problem of such a "github" would be that a scientific repository is only as good as its maintainer. In software development land we already see a lot of egos clashing over the code, but software has a clear "works/doesn't work" criteria. In many scientific areas such a criteria is hard to find and we'll have to rely on the personality of a repository maintainer.
The repositories will soon become a brand on its own. Think "nature.github.com", "physrev.github.com" and "lancet.github.com".
A second problem with this sort of company is that it often doesn't really understand the industry it's "disrupting", so they end up not solving a lot of the problems which older solutions solved.
At a more fundamental level, I think that a company is always going to be motivated to make money rather than solve a problem. Even if intentions start off well, the people in the company will believe that the company is good and needed, so they'll make choices that give up a little of the original purpose of the company to make the company survive, because if the company doesn't survive then it can't do any good, right? And after a bunch of one-degree deviations from its original direction, the company is going a completely different direction than its original purpose and isn't even geared toward solving the problem any more.
Sure, "We're going to save the world" is great marketing, but at this point I simply don't ever believe that a startup is a solution to a major problem like this.