For starters, native apps have too many permissions already.
Second, if you run a native app it should require explicit permission for internet access (perhaps at install time) so you have some idea if local data is going somewhere. Web apps are implicitly assumed to have internet access.
This should be fairly obvious, and the notion of granting more local access to arbitrary outsiders should be distressing. The fact that google and web sites feel entitled to everything on and about you computer says a lot about them. If they were humans with that lack of boundaries and sense that what's yours is theirs, we'd label then narcissists or psychopaths (not sure which).