In a JWS the header is integrity-protected by the signature if the alg isn't none. This is prominently noted in the specs and alg=none artifacts are referred to as "unsecured JWS". In a JWE the protected header is integrity protected by the AEAD cipher, because all encs must be AEAD.

The alg=none substitution issues happened because of bad usage of mediocre libraries. Other algorithm substitution can arise for the same reason. The invalid curve attacks were the ones that the spec didn't call out as a security consideration.

I support the arguments that say algorithmic agility is a bad idea and a new protocol with algorithmic agility shouldn't have come out at a time when other protocols (like TLS) were finally starting to catch on to this fact. But the JWT cat is out of the bag, and won't go back in: it's widely deployed and people are using it thinking it's solving problems they actually have. Education is the proper remedy.

The PASETO effort attempts to provide better answers and better design to an audience familiar with JWT, but there's also been an uptick in the kind of advice that heavily condemns JWT without supplying some migration paths. That latter brand of advice is harmful.