From my (outsider) perspective, I think the main roadblock atm is standardizing the component model, which would open the door to WIT translations for all the web APIs, which would then allow browsers to implement support for those worlds into browser engines directly, perhaps with some JS pollyfill during the transition. Some people really don't like how slowly component model standardization has progressed, hence all the various glue solutions, but the component model is basically just the best glue solution and it's important to get it right for all the various languages and environments they want to support.
There's a lot of prerequisites for DOM access from WASM that need to be built first before there can be usable DOM access from within WASM, and those are steadily being built and added to the WASM specification. Things like value references, typed object support, and GC support.