If you look at filtering mechanisms themselves, there are companies that try to have puzzles that they ask you to solve before joining (eg: ITA software). But they are really challenging puzzles that I would solve just for fun anyway, (which is what I did for months till I one day decided to apply for a job there). The fact that these puzzles are also made by the company (and not a third party website) makes a difference. It tells me about the people that work there and potentially about their culture. On the other hand, building a random web app to prove basic skills is not something I would imagine most talented web developers would like to do anyway.
If you look at it from the perspective of a company that is hiring, tools like this solve merely one part of the puzzle. It takes the subset of developers who are already in the market for a new job and filters them. As a startup though, what I want to do is lure the great engineers who are content at their regular jobs and not looking at all because of inertia. It sounds like there is more of a "search for developers" problem that will expand the breadth of all the developers available to me, instead of focusing on the ones who are already out there (and are most likely not the cream of the crop by definition).
In short, I don't mean to be negative, and am not saying that there might not be a market for a tool like this. But I am pretty certain that if you are a great developer, you don't have an incentive to make a web app on this.