What distro are you running? systemd-oomd kills processes a bit quicker than what came before (a couple minutes of a slow, stuttery system). Still too slow for a server you'd want to have back online as quickly as possible.
At least now when I run out of memory it kills processes that consume the most memory. A few years back it used to kill my desktop session instead!
Right, that's traditionally been because the X server has typically had a fairly large footprint, and therefore has been very attractive for the oom killer. But in the last 15 years or so, some heuristics have been applied to deliberately discourage the oom killer from killing "important things".
I install earlyoom on systems I admin. It prevents the low-memory thrashing by killing things while the system is still responsive, instead of when the system is in a state that means it'll take hours to recover.
At least now when I run out of memory it kills processes that consume the most memory. A few years back it used to kill my desktop session instead!