If you trust them to issue you a non-compromised certificate now, you should be able to trust them enough to protect your self-generated cert for a DNS-like verification and lookup system.
It's not an improvement, but it's really no worse than what we already have.
The OP for this topic, with whom you seem to be siding, was suggesting providing the CA with his private keys, and there's a massive difference between asking someone to verify your identity on a one-time basis and trusting someone to protect your identity in perpetuity. Going off of some of what was said by you and the OP...
A) I don't need to "trust them to issue [me] a non-compromised certificate", because that's not how compromised certificates work. When a CA issues you a cert, the first thing you do is check over it for errors. If there are any, you don't publish it on your site. You get a new cert. That's it. Compromise averted.
B) The primary problem with the current system is that fraudulent certificates for your site/service can be issued by unscrupulous/compromised CAs to people other than yourself. They'll be able to pose as you until you get the cert revoked, potentially compromising a subset of your users, but your idea does nothing to address that issue since false self-generated certs could be held by CAs just as easily, so we'd continue dealing with it as we do now: revoking certs, delisting CAs (e.g. CNNIC), etc..
C) Another annoyance with the current system is that if the issuing CA gets compromised, you'll probably want to procure a new cert from a different CA. Annoying, but not catastrophic...unlike the OP's idea. More on that in a moment.
D) The worst thing that can happen in any system is that your private keys get compromised. Anyone (e.g. hacker, nation state with a court order, etc.) who gets their hands on your private keys can not only pose as you or eavesdrop on your current and future encrypted communication, but can also decrypt any past communication they may have managed to collect. The current system we use has a single point of failure: you. The OP's system introduces the CA as an additional point of failure, with the end result being that when the CA gets compromised, it goes from being the nuisance it is now to being an outright catastrophe.
E) Now multiply the catastrophe by tens of thousands, since when the CA got compromised, it wasn't just you that the bad guys gained access to: they also got the same level of access to everyone else who was issued a cert by that CA.