We have always used the foam insert. I assumed it was to wick the alcohol to a higher level in the container. I am in the same situation now. Maybe I'll try it without the liner.
I was looking to buy a replacement sponge as ours have fallen apart as well. I found this same question on Thermo Fisher's website with the following response: " The white foam insert serves no purpose other than to make the printed instructions on the jar easier to read. Its presence or absence will not affect the cooling rate, and so it can be discarded. "
Haha, well, John R Carney , to make it complicated their new instructions say the exact opposite. So my interpretation, and experience with the 1 C containers (we have always used the foam inserts) is that is does not impact the freezing process. As Katie Powell mentioned, it helps to have to read the side of the jar, but I find it odd they have 2 revisions of instructions with opposite information.