Could something like mDNS be a lightweight solution to that problem?
And also I am genuinely curious how kubernetes would solve that. When you install kubernetes on all of these machines, don't you have to manually do the configuration for that either? So isn't it just choosing to rather configure kubernetes instead of your own application for each deployment? If it is that much simpler to setup a kubernetes cluster than your app, maybe the solution is to put some effort into the configuration management part of your product?
Managing many nodes is the reason orchestration software exists. Your suggestion to "put some effort into configuration management" is effectively naive, homegrown orchestration.
Build or buy? That's the same argument - except k8s is free and available from many providers.