Well, you can verify that the code that you downloaded is the same that everyone else downloaded. Even if it contains webviews.

Now if it contains webviews, it brings the security issue of... the webapps, of course.

Personally, I want an open source app. You can audit an open source app and even compile it yourself. You can't really do that with a website. And I don't mean just mobile apps, that applies to desktop apps, too. I wouldn't run a web-based terminal, for instance (do people actually do that?).