Are you saying that you have freezing weather and the water in the motor froze while the car was parked? That seems straight forward. I can't think how the water would freeze if there was sufficient antifreeze.
I suggest the "antifreeze" light would be the temperature light. If the motor got too hot for a period of time while the car is being driven then the cause may not be lack of antifreeze. There are several possible causes for that totally unrelated to antifreeze. However they can test the water and would know if there was sufficient antfreeze.
I suggest that your post isn't specific enough to answer your question. If you don't understand the tech's explanation ask him to explain again. Sincethey are going to look at the motor Iin more detail and do find a different cause then you have no worries.
I would continue working with the dealer in a cooperative manner. Do not assume you will have to pay and become defensive. As to rights, as others have said if you've had regular and on time service performed by the dealer they will have a record.
However, it sounds like they are saying the car was driven while too hot and the driver should have seen the too hot light come on and turned the motor off. That has happened to me. If that is the case they can test the light to see if it's working. There are many reasons the motor heated up unrelated to antfreeze. If the issue is the "idiot" light what has to be determined is if it's working. If it's working when tested I don't know of any way to prove it didn't come on.
I would've known something was wrong even if the light didn't come on. The motor smelled hot. How long was the car driven once it was emitting smoke? The driver should have immediately pulled off the road and turned the motor off.
You have the right to get a second opinion. You can also appeal. The dealership and/or the insurance company can tell you how to do that.
From experience, I suggest you first work with the dealership. It is the insurance company that will pay. Unless the dealership was at fault in not providing adequate service they have nothing to lose by being on your side.
I suppose that the starter could be related if it was damaged by an electrical problem. The mechanic could give you more information. I suggest you need to talk to them with an open mind. Take notes so you can understand how everything is related.