Not only does this threaten the separation of church and state, since religious schools are of course included, but it also siphons money directly from public to private institutions. In doing so, it secondarily rewards the rich, because who is likely to contribute to a scholarship fund in the first place?
Here is a map showing the states that have a similar plan right now. Not a lot of blue states there, right? But a lot of early primary states, right? Interesting.