What does he mean by "I do usually try to monitor them with a dead man’s switch.", when talking about backups?

Dead-man's switch means triggering when something doesn't happen. (The name comes from a switch that an alive operator would need to hold in such a way that if they died they would stop holding.) So in this case she means that her monitoring will fire if there wasn't a successful backup within some configured period of time.

I assume this is opposed to alerting when the backup job fails, which is an issue if the job never runs, or hangs forever, or crashes in a way that doesn't trigger your monitoring.

However I don't see how any of this solves the issue of not testing your backup. Because you can definitely have a backup task succeed regularly but the thing it is backing up is still unusable.

https://en.wikipedia.org/wiki/Dead_man%27s_switch

In this context it means upon a successful backup, update a timestamp somewhere. Some other system monitors the timestamp and if it ever becomes more than for example 1 day ago, it fires an alert.

I don't call it a "dead man's switch", but I absolutely monitor some directories for new newest file. If the backup monitoring script doesn't find a file less than 24-ish hours old in the backup destination directory at any time it should send me an alert.