This doesn't really make sense, given that the wasm used by websites is going to import a bunch of JS functions as dependencies. You're not going to have those available in any native environment.
> Is it really the case that browsers have default-enabled all sorts of extensions that are not yet widely supported by the rest of the ecosystem?
Yes
Photoshop in particular is a good example of a bleeding edge wasm app - browsers had to relax restrictions on things like function pointer table size in order for it to work. So I wouldn't expect it to build and run anywhere outside of v8 or spidermonkey.