> Why wouldn't I be able to run GTK apps on their ICE powered desktop?

Of course you would be able to, just don't expect the theming to match, the menus to have similar structures, dialogs having buttons in predictable places, etc.

> isn't the purpose of WASM to make possible in web environments that which is possible on desktop environments?

I don't think so. I think this is the purpose of all the new Web APIs more than WASM. The sandboxing and the inability to call out to the system is crippling for GUIs in WebAssembly. It works well if you control the drawing and are able to output raw data to a <canvas>, but it's a hassle. WASM works the best for things like opencv or dlib, but I think the GUI for the apps compiled to WASM will need to be written in JS for quite some time, if not forever.