How could the specific heat of the substance your container will hold affect the performance of the container?

I am not entirely sure, but I hope this helps ;)

Heat transfers from a very hot body always, and into a colder body. This is due to the change in temperatures at which the two bodies exist, and also it depends upon its magnitude. And, the heat exchange stop once both the bodies attain same temperature.

And from the relation of heat equation, we get that it is directly proportionate to the substance's specific heat. But, there is no such thing that the area of contact is important, for heat transfer. It gets transferred although a small point is in contact.