If a private domain and/or the server is only reachable from private IPs, then its IP should be in an asset management system somewhere (it should be if it is public too); be that a full blown orchestration system, if cloud servers or containers, or some form of asset register - even if only a spreadsheet - if it is physical servers in your own data centers.
If properly automated, that asset register should drive configuration deployment - you shouldn't ssh to it to fix a broken dns configuration, you should run your deployment with a fixed config.
I get that this is an idealized setup and that a lot of the time things intervenes and it's quicker and easier to do things manually then and there. Been there, done that, but in the long run it pays to invest the time in then cleaning that up (up to and including wiping clean setups that have been manually updated to verify that the automated setups are correct and complete).