I said nothing about making it public, because I don't view the possibility of someone capturing DNS packet of the kind "A? fooserv34.hide.example.com" as making anything public. (And even if your authoritative DNS server would allow AXFR for whole hide.example.com you would only leak information that is useful for somebody who also have capability to find out the exactly same information on his own).

The convenience is about not having whatever VPN solution you use to inject DNS recursors into laptop's OS, which end ups being not reasonably solvable when you have two such VPNs you have to use at once.

Who said anything about making IP addresses public? You seem to be assuming registering a domain name implies delegating that domain to an actual name server that must respond to public requests. These assumptions are both incorrect.

Registering a domain prevents those names from being used for other purposes in the future. Your domain does not have to have valid NS addresses. Even if a domain has NS record with a valid address of the authoritative name server, that server does not have to respond to requests from the public internet. It doesn't even have to have to be accessible on the public internet.

Register a domain for internal use and run an internal name server on e.g. 10.x.y.z that handles your private IP space, and configure the local recursive resolver name servers to hand out your internal server's address when asked about the associated NS record. At the same time, set the real NS records for your internal domain to e.g. a traditional DNS hosting service that returns a CNAME pointing to your public domain. (the public internet only sees *.private.example.com as a CNAME to www-public-name-com.example.com)