WASM simply is a better compilation target. More general purpose, easier to optimize, more compact, can load incrementally, etc. If you are going to compile your code anyway, you might as well use a proper compilation target if you have the choice.
More generally, I think WASM is starting to challenge the whole "there can only be DOM/CSS/JS" thing that has held web developers back for two decades. It was fine when that was the only thing that worked but there's a reason mobile developers, game developers, etc. prefer native: the web sucks. There has always been this creative tension between what designers want and what the web can do. We used to have things like Flash, Shockwave, and Silverlight. HTML 5 has largely failed to replace those. We deprecated the plugins for security reasons. But we never got a decent replacement.
With WASM there's no good reason left to keep on preferring DOM/CSS/JS. It's there if you want to, but there are some alternatives now. In its current state, that still requires things like C++ and Rust.
But WASM is maturing rapidly. There are a bunch of things behind feature flags in browsers (like the new GC stuff) that will level the playing field once those flags get removed. From there, it's mainly a matter of UI frameworks and tool chains that need adapting. Jetbrains is actually targeting wasm with compose multi platform. It renders to a canvas. Currently experimental and it only works if you set some feature flags on your browser. But it allows for portable UI development with Compose and allows you to target Android, IOS (alpha release), Desktop (jvm), and soon Web via wasm with the same codebase. There are probably other UI stacks that might soon start targeting web. Blazor of course was an early adopter but it ships its own GC and that makes for a bit chunky apps. And of course there is a wasm based replacement for Flash.
HTML 5 is no longer the only game in town. It will have to compete on merit now instead of merely holding a de-facto monopoly.