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Chrome doesn't even work with its own company's social network plugin (Hangouts). I always have to switch to Firefox to use screen share.

I haven't encountered problems with Hangouts. But the new Firefox Hello service for in browser webcam chats is pretty awesome: https://support.mozilla.org/en-US/kb/firefox-hello-make-rece...

(disclaimer: I do actually work for mozilla)

Seems only tangential, since I don't see anything about group calls or screen share (which are the main features we need Hangouts for).

I believe group calls will be in Firefox 35. Not sure about screen sharing, but it's being worked on and coming soon.

That may be true for you, but I assure you that is not generally the case.