That might be a good idea as a last-resort measure, but if you're only finding out problems because customers are telling you about them (even indirectly through a signal like this), your monitoring and alerting is woefully inadequate.
Can't tell you how many times I've had a customer write to me, only to be receiving some automated alerts 30-40 seconds later.
But as a counterpoint I was once working in a project where our monitoring dashboard showed an anomaly in incoming traffic. It turned out that it was an ISP problem and that we were the first ones to notice according to them.
So maybe part of the answer as to why customers are faster is that different monitoring systems are monitoring each other :)