the main dev of tox is kinda toxic to it's own project with kneejerk responses like these: https://github.com/TokTok/c-toxcore/issues/426 and https://github.com/irungentoo/toxcore/issues/121 showing much more enthusiasm than competence when it comes to security and crypto. while the people behind briar i got to know as reasonable and deliberate people.
can't say much about ring, but a few years back i looked at the crypto and it was boring, which some people argue is good, i was not sure about that in this case. dunno how much they evolved since then.