I actually don't see a problem with using a document renderer for applications, assuming it's actually treated that way. The document-rendering of a browser is lower-level than a UI toolkit, and that makes it more powerful in a lot of ways. But we need to be able to build abstractions on top of it to actually build UIs. I think react.js's approach (or something similar) is actually pretty good here.

A lot of the problems we run into are people treating DOM nodes as if they were UI widgets, when in reality they are lower-level primitives for drawing and capturing input.

There's canvas, build your own UI toolkit on top of that if you don't like the DOM.

Well, it's not suggested to stop with WebAssembly, it's just aт evolutionary step from the situation formed to this time.