It seems reasonable that if they had to go through the stores on their way back to the brood and cluster they might be more inclined to get a snack and bring some food up to the cluster while on their way.
It does not go that way.
Our bees do not fly during winter. If they fly, they cannot return.
System is such that cluster starts wintering in the place were were last brood.
Cluster makes heat which rises up.
Bees eate stores empty downstairs before they move slowly up.
In deep frost bees cannot go down because it is so cold there. But they can move up.
When there is mild weathers, cluster widden and can reorganize again and it makes their movements and reach the food edges.
in deep frost they are in slices between combs. If the gap has no honey, that gap of bees will die. It cannot move though the comb.
So bees move over the top bar when cluster becomes smaller.
Bigger losses happens when deep frost lasts long time, like weeks.
A big colony can move easily because it generates heat into whole box. It reorganize easily itself and reach food stores.
I have seen now that many hives do not start downstairs but it start near entrance and near front wall. The cluster peeps in uppper and lower entrances. The last winter food stores are near back wall in upper parts.
.The system is finally that at the beginning of spring build up the brood has warmest place in the hive and it is up.
It is the best way to save winter food (tree trunk)