Not sure how you define a prior shelter that isn't intact.
Certainly ice could be stored near a source.
Insulating it was the priority for storage.
Doesn't matter to the ice if a home was near.
They are gone now but I've seen some pretty danged small shacks people slept in.
Material from a railroad about a half mile away some of them.
At first I thought they were for a pair of mules or horses if they wanted shelter.
(They don't use knobbed doors though.)
There were shelters elsewhere . Multiple elsewheres . Barely big enough to stretch out in but isn't that enough when ones work is outside?
Some were remote enough that especially in winter they could be vital if caught in rough weather.
Most had an outhouse or reason for the structure. A water site where summer things could be cooled. (Creeks usually.)
One a river sight where water craft could be secured in a dug trench .
If time nd luxery allowed just covering a spring kept them cleaner.
Enclosing one is similar to having a spring house.
Some areas on shores or near them saw pic nics and other get togethers in the summer or other seasons.
Back when...When you arrived at a gathering site your horses or other draft animals needed watered. Or should have been depending on conditions and distance and your care of them. That meant water nearby. But not the same water you drank and it didn't need to be cold.
A roof could be a bonus over human drinking water at spring or artesian well.. As of course a spring house could be. As kids we had a roof over an artesian in the back yard. Always cold feet when under it in summer.
Absent springs in the area to cool food in warm weather , we're looking at your ice house concept!
