Forking HTML might not make sense indeed, but I think this post doesn't make a compelling point as to why or why not.
If you step back a little what we have today is a combination of a common browser implementations (aka. the DOM tree + render logic) and bunch of standardized parsing and interpretation logic that maps textual versions of CSS and HTML to this. For better or for worse, this stuff is standardized via WhatWG & W3C as well as the three remaining browser engines (chromium, gecko, and safari) whose intersection of behavior…
If it is so hard to layer a coherent and not laughably overcomplicated runtime on top of a document viewer in 20 years, then it follows that it probably wasn't such a good idea to begin with. Unless you're hellbent to freeride on the web's success, and ruining it in the process. At which point we're trying to solve an economic problem (someone else's) via a technical solution.
To get a sense how far of the mark browsers still are, you only have to look into the problem of rich text editing on browsers, which is a natural step up from pure text browsing, and something the first browser did already. Today, this is only barely possible thanks to a couple rich text editor projects who made heroic efforts to work out browser quirks (with contenteditable such that the browser's spellchecking can be leveraged, or alternatively using Canvas/WebGl, essentially developing a browser-in-browser).
In a situation where browser vendors struggle to keep up, adding additional tech such as WASM is the last thing you want to do. When and if WASM gets even basic language infrastructure such as gc, DOM or WebGl bindings, or anything even mildly interesting, then it'll immediately become another maintenance problem. And as you said yourself, at best WASM can help port apps over to a browser. It's however not clear what the purpose of that exercise should be when said apps run just fine outside the browser. I'd say if there were indeed so many apps waiting to be ported over to running in the browser, then there have been sufficiently sophisticated transpilers for like 15 years now (emscripten, gwt, many others), so I'm not buying that as an argument to make browsers even more complex.