In the distant past there was plenty of times I tried to fix dying web servers by logging in and restarting a rogue process or cleaning up a memory leak... only to have my SSH session and shell sessions being repeatedly killed by OOMs.

Not as big of a problem as it used to be. Both because OOM killer has gotten better and also web servers are such a commodity that remotely cycling them isn't going to cause a outage and is the quickest and cleanest way to get them back into production.