We had a server that went down and didn’t generate alerts for Heartbeat Failure or Could not Connect to Computer.
What I’d found is that there is a group in SCOM called
“Managed Computer Client Health Service Watcher Group” and there is a default
override to disable generating alerts for Heartbeat Failure or Could not
Connect to Computer against this group.
This group is apparently intended for workstations being monitored by SCOM and is dynamically populated but sometimes servers also ended up in there.
I you don’t monitor workstations the easiest solution is to create a second override to enable those alerts and just enforce it.