I personally think NIST should be disregarded, but you can disregard NIST and still end up with CRYSTALS-KYBER as your default PQC KEM, on its own merits, which can include the fact that NIST's standardization spurs so much implementation of CRYSTALS-KYBER that it becomes a de facto standard in addition to a de jure standard. (Same for signatures, and so on).
People with qualms about NIST might also reasonably have qualms about AES. And there is a common cipher that people use outside of AES --- Chapoly. But it would be downright weird to use, like, Serpent or Twofish; it would be the cryptography equivalent of a "code smell". Chapoly and AES are the de facto standards, and OpenSSH supports both.
Again though: my question is just, what does this (frankly weird) Bernstein complaint have to do with any of it? Bernstein himself is a NISTPQC participant; he's on one of the (large) winning signature teams.
(I think all the technical details here are super interesting, but not especially motivating; I'm not a cryptographer and you should disregard me as well, but my basic take on QC crypto attacks is "Rodents of unusual size? I don't think they exist.")