What about if you have private DNS servers that has sites that cloudflare does not have? For example internal intranets etc? So mozilla will not work at all in that case?
See the similar problem with TLS certificates...
(edit) Ok, that was indeed put more dramatically than necessary. My point is that private DNS names seem to be heavily discouraged by browsers default configurations.
You can change both the DNS resolver as well as install custom CAs - however, this has to be done again for each client. If you want to have your sites visited by clients that you don't administrate, you're out of luck.
(warning - rant follows) The direction browser vendors would like the ecosystem to move also seems quite clear to me - there is a strong push to get everyone on HTTPS, at the same time CAs are themselves increasingly regulated by browser vendors and cannot hand out certificates for IPs and private DNS names anymore. Now the next step seems to be DNS. If that is not a platformisation of the web, I don't know what is.