Seems like you have described apps for snoop-phones. Those are a refresh-less, not device-agnostic browser apps with no tabs, no ability to change a font-size and limited ability to block ads.
Ah, but that's not that case here at all. It's not an app. It's still a sandboxed web page. The only difference is the communication method. Rather than a transactional client-server relationship, we adopt the media-steaming approach post-load.
That is to say: You would load your desired website (the standard way) and then experience a "refresh-less" session for the duration of your visit.
Imagine visiting this website, Hacker News, to read a few posts/discussions. Does the idea of circumventing a refresh offend you? You could click into a threaded discussion (from the homepage) and back without triggering a reload or needing the data beforehand.
Other than the indexability problem, it seems like an ideal solution. In many use cases, the indexability problem is actually a feature too.