> The only time an employee needed to find their own cover is if something unexpected came up during their on-call period and they needed a few hours out (like an emergency visit to the doctor with a child etc).

That's exactly the time where "finding your own cover" is the most stressful.

I do see your point, but I think some context about where I work helps here. It's a very chilled company and all teams are very self-organising. In the case of me needing to find cover, I would just drop a message in our on-call slack channel and someone will almost always pick it up (and do the necessary stuff like adding an override in the opsgenie rotation). If nobody happens to see it (unlikely) then I would just let the alerts escalate to the next person who would be happy to pick it up because they know that people don't let alerts escalate without a good reason. Everyone cares about their colleagues so they want to help if they can (it's also quite a small company).