I'm excited that we developed the same thing so close together. I'd be curious what your inspiration was, because maybe it was similar :) We found an article that showed a grandma facetiming to play chess.
We'd recommend checking out Janus webrtc. It worked well for us, and will work on mobile/firefox. We've used it for years, and it's well developed. Video rooms up to 6 people too, so you could have more options for games