The starter is only used to turn the motor. If the car is not starting correctly when warm, take it to the dealer and complain. IMO it is unlikely you have a bad crank sensor and again IMO yes, you'd get a CEL if that were the case. Don't be surprised if the dealer says "normal." I don't know whether you should be dismayed in that case or not, but I've not heard of anyone with the same complaint. When you say it takes long to start, how many cranks are we talking about before the motor catches? 3 seconds? More?