For example, consider a web-based mail client. You want to send an encrypted message, say via PGP, and you don't want the server to be able to read it, even if the server is evil. You'd like to be able to do the PGP encryption 100% in-browser, with no browser plugins or extensions necessary.
I think that's the most common category of use-case for browser crypto. Unfortunately, it's one where browser crypto plainly doesn't work. The whole point here is to defend against an evil server, but if the server is evil, it will send you evil crypto JS. TLS doesn't help you. Nobody's impersonating the server or altering the JS file in transit. You're getting an authentic copy of the JS file from the real server. It just happens to be an authentic copy of an evil JS file.
Given that, what can you do with browser crypto, practically speaking?