1. It has the ill-conceived "alg": "none". But this feature made breaking JWT so easy and low stakes, that many libraries have removed this feature completely, or just disabled it by default.
Modern RFCs that mandate JWT use in servers (e.g. RFC 9068) often explicitly forbid this, and the latest BCP for JWT (RFC 8725) recommends that libraries only accept or generate tokens with "none" when the user _explicitly_ requests that. And yet, we're still seeing "alg": "none" vulnerabilities even to this day. I'm not sure if it made sense to support "alg": "none" in the first place, but if we ended up doing that, the RFC should have been much more strict about this.
2. The other issue is mixing up asymmetric and symmetric encryption. You can't embed the HMAC password directly in the user-generated message; but if the library is not built securely, it would just treat the public key itself as the HMAC key when it gets an HMAC alg in the header. This makes forging tokens quite trivial if the library is misconfigured.
JWT is not nearly as bad SAML and its designers learned some important lessons (simpler base format, no canonicalization or embedded signatures), but this is still a design-by-committee standard that didn't properly involve. The full JOSE standard (including JWA) is even worse, but fortunately JWA doesn't get used a lot.