If you're talking about a game server like this, though, where you don't have sandboxes running client-originated code, just processes running your own software

If you're talking about a game server like this, though, where you don't have sandboxes running client-originated code, just processes running your own software, any remote code execution vulnerability is already going to be in a position to do incredible damage. There are almost certainly going to be other privilege escalations available to you once you've buffer-overflowed or whatever to dupe the server into running code you control in process - and honestly, once you own the main process on a server like this you can probably do enough damage from in there without even needing to escalate. I think the frightening thing about meltdown/spectre is the ability for code to escape otherwise solid sandboxes - including VMs, as well as javascript runtimes. It's a change in the threat landscape for systems which expect to run remote code but believe they can do so safely.