Aside from security, you might face unexpected behavior after long time it receive last update. For example,
- Repository or website which host software for your OS no longer exist.
- Some application cannot use internet connection properly due to certificate problem.
Yes, official repos may go down after some time. But like in case of Centos 6 there are some archive or even 3rd party repos you can continue using.
I'm mostly using systems with no GUI but I'm yet to see an OS which can't connect to the internet because it's too old. Until recently I was running a Windows 2008R2 machine which was complaining all the time and the browser stopped updating but you could connect anyway.
What exactly do you mean by rock solid? Rarely crash?
Yeah, that, and also security-wise. Firewall keeps working, kernel has no critical vulnerabilities. Ar least they weren't exploited. Of course, I wouldn't run something connected with finance, payment processing, health, military or some other sensitive data on such a machine.