This isn't a batch scheduling problem. It's a monitoring and alerting problem. OP needs a monitoring and alerting solution.

Stop ~~~ engineering ~~~ things and choose the right tool for the job.

"Solution" and "Job" may be overstating things here. If all op needs is monitoring and alerting of a couple of machines, then Datadog/New Relic/Ops Genie/Pager Duty/Server Density/Pingdom/AppDynamics/Loupe/Sysdig/Dynatrace (just to name a few in the crowded space) are all likely overkill and not worth the cost.

A large portion of the cost of many of these tools is spent "choosing the right tool for the job" Figuring out what they do, what they do well, where they overlap, where the company that makes the tool is headed, how hard its going to be to swap that tool out for a better one (or cheaper one) - thats a lot of expense in labor and training.

A script scheduled to test something and notify? Hardly ”engineered”.

I claim using pretty much any tool other than jenkins is easier to maintain over time (as illustrated in this very article).