Public name server (controlled by me, I own example.com):
private.example.com. 300 IN CNAME myserver.home.
Private name server (some auto-config voodoo, can't modify this record):
myserver.home. 300 IN A 192.168.6.66
But private.example.com only works intermittently: It did use to work in Chrome, but currently doesn't. dig without further options returns the cname, but dig with +norecurse fails, each tested with the local dns server and several public ones.
What is the easiest way to fix this? Something that doesn't involve setting up a local nameserver, preferably?
Background: In my home network there is a server "myserver.home" which is accessible as "myserver.home", but this is obviously no valid public dns name. It does work fine in the home network, though. Because I want HTTPS, I added a CNAME from private.example.com to myserver.home so I can get a certificate for private.example.com (using certbot, dns-01, works just fine).