It can already interact with the dom and people are working on making this more seamless and faster.

The real question would be: why would you do that when you can instead use any proper native UI toolkit?

Answer: it's now a choice. If you need command line stuff like vi running smoothly in a browser, the DOM is an obstacle, not a solution. If on the other hand you want a wasm application driving the DOM and generally looking and feeling like a web app, you can do that as well. There are several frameworks emerging for rust, kotlin, C#, and other languages that do exactly that and that use bindings for the DOM and other APIs you have in javascript in a browser. Basically, you are writing normal dom/html/css based applications; except it all compiles down to wasm and does not involve any javascript.

Well, HTML/CSS are sort of separate and wasm doesn’t replace them, but if you mean DOM access, it’s planned but not available yet. You basically have to write JS glue code that interacts with your web assembly code