It has the potential to do so, but I'm waiting to see it for real before calling it. Right now it's just not the case.

I'm not sure how. WebAssembly is more limited than JavaScript (e.g. no access to DOM), and also more limited than native code (e.g. much more limited threading story).

I wonder how long after the WebAssembly DOM API becomes stable tbe first Rust-based "server and client written in one language" frameworks start popping up.

I thought they were working on adding DOM stuff too?

They are. And they are also working on having full multi-threading (thanks to SharedArrayBuffer).