I definitely agree that we
should be improving browsers in that way, though I'm not sure that it's easy and I'm pessimistic it'll ever happen. Mostly because - cynically - I think that the vast majority of users will never care about owning data and software, so I don't see the incentive for browsers vendors to implement features to make it possible.
In their current state, I personally don't find PWAs + local databases ergonomic enough. So actually what I'm hoping to get out of the platform I'm building is - somewhat egotistically - an alternative that works for me personally and possibly for a few others like-minded individuals, not a browser replacement to take over the world.