Theory and practice doesn't match in this case, and many people have remarked that companies that sit on the WhatWG board have vested interest in making sure their lucrative app stores are not threatened by a platform that can run any app just as well.
I remember when Native Client came to the scene and allowed people to compile complex native apps to the web that run at like 95% of native speed. While it was in many ways an inelegant solution, it worked better than WebAssembly does today.
Another one of WebAssembly's killer features was supposed to be native web integration. How JS engines work is that you have an IDL that describes the interface of JS classes which is then used to generate code to bind to underlying C++ implementations. You could probably bind those to Webassembly just as well.
I don't think a cross-platform as in cross CPU arch matters that much, if you meant 'runs on everything' then I concur.
Also the dirty secret of WebAssembly is that it's not really faster than JS.