I'm not sure how email would help. I guess you mean the system would email you once a day saying that the monitoring system is working and if you don't see the email you know to check into it. The monitoring system I use has a higher SLA than 24 hours.
Usually folks divide the monitoring work among two servers and each server monitors the other. Or, you "meta monitor"... a monitoring system that just monitors the monitoring system. Then you get a third-party to monitor that. Then it is turtles all the way down.
It sounds like both of you are talking about cloud stuff when thinking about datacenters. I can see how that many layers and cross-checks would be necessary when all you really control is running memory and some pieces of storage, but a lot of that is due to the platform. When you control the actual metal, third-party monitoring services are much less necessary.
For a real DC, when it goes down I get a phone call from a human. I don't have to reinvent that process. If it's my own server room, I use a landline and a modem for OOB "dude you gotta come down here" notifications, a WAV of Woody Woodpecker or something. If the phone lines are down, I look at the newspaper headlines to see what happened.
There's no reason not to set up a standalone monitoring regime. Whether or not you use heartbeat notifications to tell you all is well is a matter of taste, but there is definitely more to maintaining your nines on a daily basis than simply adding more layers of monitoring.
Usually folks divide the monitoring work among two servers and each server monitors the other. Or, you "meta monitor"... a monitoring system that just monitors the monitoring system. Then you get a third-party to monitor that. Then it is turtles all the way down.