If you're going to talk about vulnerabilities that you don't know about yet, then you should be looking at how you're running old kernel versions and unpatched software. Hiding your hostname doesn't protect you against anything other than insecure application code. Old httpd or openssl will still be exposed.

There are plenty of ways that your secret hostname can leak. It's just a weak, low-entropy, guessable password that can't really be rotated. And moreover, if you're going to choose one thing to do (or time box your efforts around security), it's the highest complexity I can think of for almost no real return. I mean, if you're so concerned about leaking the hostname from your HTTPS cert because your site is that insecure, why are you even using HTTPS? What are you even protecting?

Is that the best way to spend an hour securing your site? Even just slapping the six lines to put hard coded HTTP auth credentials on your nginx virtual host will do more than that.

Require TLS Client Certificates at the load balancer level (or even just HTTP basic auth).