I did have an old laptop with bad SSD that would have similar problem - cannot ssh, but can ping, and desktop quickly gets unresponsive. On root device issues kernel remounts root drive to read-only, and that breaks everything, but the machine still works - yet you get no logs because /var/log is also read-only. I did some googling and it seems SSH requires writable /tmp, so you might setup some in-memory one, if you haven’t already - you could check if your machine uses tmpfs (in-memory tmp) to reject this hypothesis.