Azure Traffic Manager monitors its endpoints using Endpoint Monitoring. I am unsure what you mean by 'status monitor', but Traffic Manager will monitor the endpoint to check for outages. If only one instance of many goes down on your endpoint, it should still be responding, and the endpoint will remain up.
For Example, your 'site1.company.com' is hosted on an Application Gateway and one of your instances goes down, the Application Gateway should detect and stop routing traffic to that instance, but the endpoint 'site1.company.com' should still be responsive and distributing traffic.
If all instances go down, then the endpoint will report as unhealthy, and traffic will not be directed to that endpoint.
If all of your sites are hosted in the same place, and you need to fallover to a backup site, you can still use endpoint monitoring with priority routing to send traffic to your DR site in the event your original site becomes unavailable. Traffic will only go to your DR site in the event that your original endpoint becomes unavailable.