Perhaps it seems obvious to some, but it's not obvious to me so I need to ask: What's the advantage of a selectively-available DNS for kids playing Minecraft with Nintendo Switch instead of regular DNS [whether self-hosted or not]?

All I can think of is that it adds obscurity, in that it makes the address of the Minecraft server more difficult to discover or guess (and thus keeps everything a bit more private/griefing-resistant while still letting kids play the game together).

And AXFR zone transfers are one way that DNS addresses leak. (AXFR is a feature, not a bug.)

As a potential solution:

You can set up DNS that resolves the magic hardcoded Minecraft server name (whatever that is) to the address of your choosing, and that has AXFR disabled. In this way, nobody will be able to discover the game server's address unless they ask that particular DNS server for the address of that particular name.

It's not airtight (obscurity never is), but it's probably fine. It increases the size of the haystack.

(Or... Lacking VPN, you can whitelist only the networks that the kids use to play from. But in my experience with whitelisting, the juice isn't worth the squeeze in a world of uncontrollably-dynamic IP addresses. All someone wants to do is play the game/access the server/whatever Right Now, but the WAN address has changed so that doesn't work until they get someone's attention and wait for them to make time to update the whitelist. By the time this happens, Right Now is in the past. Whitelisting generally seems antithetical towards getting things done in a casual fashion.)

Why do you want to do this? What would you redirect / override on this?