Does EGR monitor readiness have something to do with a possible faulty coolant temperature sensor or anything cooling system related?

is the check engine light on also? P0401?

Trying to get monitors to pass can be frustrating. Although you didn't really mention if your getting a check engine light or not. Some monitors are specific about the enabling criteria for starting and completing their tests. Most do include engine temperature. So , If your engine is not reaching operating temp then it could effect certain monitors passing or not. Here at Street Tech Auto Care we have a great record of getting them through the monitors and would be honored to help, but if you are trying to do it yourself then investigate the specific criteria for the specific monitor that wont pass and concentrate on getting it to do those things.