The Unreal 3 demo using Flash is still on YouTube.
And this is why most game studios are playing wait-and-see with streaming instead, proper native 3D APIs, with easier to debug tooling (Web still hasn't anything better than SpectorJS), and big size assets.
I don't think it would have the same issues, because the files could be stored in a user specified location outside the browser's own storage area.
Browser vendors can't just delete stuff that may be used by other software on a user's system. And they cannot put a cap on it either, because users can store whatever they like in those directories, bypassing the browser entirely.
But I have never used this API, so maybe I misunderstand how it's supposed to work.
So the web app could ask the user to pick a directory for model storage and henceforth store and load models from there without further interaction.