How are users supposed to know whether they can trust a desktop app with near-unlimited access to their computer? Generally they don't, but blindly install it anyway because it seems to do something they need.

Yes, that's called "opt in" and it's the same security model this standard is proposing.

And your local apps don't self-modify the code all the time, whereas that's exactly what the dynamic browser environment with 3rd-party scripting is.