You could certainly use zero knowledge proofs to prove some statements about the secret key, like requiring it to be roughly a 1:1 ratio of 0s and 1s and the longest run of the same bit is not significantly larger than log2(n). I think if I were trying to enforce a minimum level of security then I might take that route, requiring that the key passes zero-knowledge proofs of various statistical tests of randomness.

By pigeonhole principle, nearly all numbers have to have a Kolmogrov complexity within some constant of their own length, meaning they are algorithmically random and will by definition pass every statistical test of randomness, so you lose surprisingly few bits entropy per assertion this way.

You're looking for https://www.ietf.org/archive/id/draft-ietf-lamps-csr-attesta.... It's gotten enough renewed attention in the wake of the eMudhra fiasco that I'm reasonably optimistic about it becoming required for intermediates.

Do you know how many SSL certificates are issued per year? LE issued 10 million per day in 2025. https://letsencrypt.org/2025/12/09/10-years

And again, you have to actively work to get a "private key" of 0000 ... 0-255. If you want to be stupid it's not really the CA's job to tell you not to be. It's their job to verify your identity. And, again, a CA itself is a totally separate beast where this thinking does not apply. If you want to be secure it's easy to be secure and if you don't want to be... Why should anyone else care? You are only hurting yourself.