I think you can't just split. I think you need to be able to nest both. For example a lot of sites will want a styled header for "branding" and will refuse to use "document mode" if they can't have that, so they will render their whole site in "app mode" which is basically the same today. So I think you need to let them render their header in "app mode" and the text in "document mode" and even embed a little game/tweet/calculator that is in "app mode" inside the document.

Basically your last paragraph. I think if we want to let people use the document mode we can't lock them into a restricted environment. Instead provide them a better user experience that they can opt out of at any point when they need to. Otherwise the first limitation and they will throw the whole thing away.

Unfortunately this means that scripting will still be available everywhere, but I actually think that is fine. I know scripting can be used for evil but it also can produce great experiences when applied lightly in the correct places.