For chest shape chillers/freezers, open at the top, the cold will tend to stay in, as cold air sinks, hot air rises. If they have a closed lid then every time they are opened, cold is sucked out and will be whooshed into the air. Without a lid, there will be less perturbation of the air when goods are removed. The problem is the upright ones, where the cold can fall out of the bottom if there is no door. However, every time the door is opened the same thing will happen, plus the air being whooshed out by the sucking motion of the door. I don't think I've seen vertical chillers/freezers with out doors - maybe I'm just unobservant.
To my mind, the best option would be a chest type cabinet, with a sliding top - no air is sucked out, and the lid can be closed gently.