I've had this same issue over and over again in my career.

I've toyed with the idea of writing a daily "sanity checker" in crontab that verifies various concepts of system health.

Examples: Did the latest batch of data transfer to S3? Did we delete old customer accounts today? Did we get any signups (because if not, something may be broken, but not triggering an exception report etc)? Did we send out daily report emails?

But I could see this easily becoming a pointless exercise, and I doubt I'd have the time to keep the sanity checker updated with the latest requirements. In fact, the sanity checker would probably become insane pretty quickly.

Perhaps the platform itself should do this for you, in some way. Idea: while coding, indicate that this procedure should be running periodically, ie:

    Monitor.registerPeriodicTask('email-reports', 'daily')

and then the system would log every time it occurs, with a generic task that would run periodically and scan for things that should have occurred, but haven't in some while.

Monitor that the newest backup is less than N hours old.

This is true, without the right tools. I am moving to logstash with kibana to do this, and it's looking very promising. See http://www.elasticsearch.org/videos/kibana-logstash/

This was true before Splunk. If you logged too much, your logs could start to outstrip the assumptions behind your log rotations and cause trouble. Now the common wisdom is to just log everything so you can Splunk it later if you have a problem. Verbose logging + Splunk have made production incident identification so much easier than it used to be.

Splunk DOES charge by the GB, but it's not very expensive in the long run.

My favorite systems to work with are the ones with overly verbose logs, where the overly verbose parts were clearly tagged and could be filtered out. Generally, we would never look at the verbose lines, and even when we did, we would normally have some idea what we were looking for, and be able to filter somewhat for it.