In most cases, the sensor will indeed set a check engine lamp. Typically
the failure mode is an open circuit, which the computer easily detects.
If, on the other hand, the sensor is just off 10 to 20 degrees or so, it's
much more difficult for the ECM to pick that up.
I don't thing the coolant temperature sensor is a high probability item.
It's just one of the things that should be considered.