Old world Ice Houses were insulated, usually with
straw and sawdust was put on and around the ice blocks.
The ice was harvested from frozen lakes with large saws and then loaded onto wagons, this was pulled to the ice storage "cave" and there they were stacked and dusted.
I think what you are missing is that the very nature of the old world ice houses prevented water inside to freeze.
Wine cellars are mostly the same as the old ice storage spaces, they remain a constant temperature, summer and winter.
If you have properly constructed an ice cellar, the outside cold
should not penetrate
enough to freeze water in a container.
You would have better success if you did your freezing out side then moved the frozen containers inside.