One option is to send cron script execution status to Zabbix (and alert on non-zero status and on nodata). But using Jenkins sounds interesting. This article does not list any details of your implementation, though.

Sending itself may fail. That's the reason to use an external job control system such as Jenkins.
As for implementation, there's nothing specific, really. Regular Jenkins jobs.

I mentioned that when sending fails you get a no data alert after a while :) but it's not ideal, of course.

Yes, right. Still, it requires additional trigger configuration. And of course, Zabbix itself.
Jenkins provides much more visibility. (Although, any other CI system should do just as well).