Well, after the air exchange: the fridge is filled with warm air.
You have a point if the fridge is full with stuff that is cold, and does not bother about that "little bit of air".
But if the fridge is only half full or even only 1/10th ... three or four openings and the stuff is warm inside.
Home batteries can only deliver 800W to the house (assuming a "Balkonkraftwerk"). But they usually have a second off grid plug, which can deliver 2kW or more. The intelligent switch over would be needed on the fridge side: take power from the off grid side of your system, or from the house. Not sure how that is done in reality.
A "standard on grid" installation would power the house with full power, not the mentioned 800W. Then everything is just normally attached to the house plugs and no machine knows where the power is coming from.