Only heard of seed being stored that way, but that is tougher than the plants. It may be a case of minimum temperature before they actually start marking. I believe it is basically down to ice crystals forming in the sap why they actually get damaged. I would not try putting any plant in the fridge you are not prepared to loose.
https://cactuscare.com/survive-freeze/
viewtopic.php?t=9782
https://homeguides.sfgate.com/lowest-te ... 92007.html
Kew Gardens does have a "refrigerated" greenhouse for alpines, though not "refrigerated" in the usual manner:-
"The new house features a set of automatically operated blinds that prevent it overheating when the sun is too hot for the plants together with a system that blows a continuous stream of cool air over the plants. The main design aim of the house is to allow maximum light transmission. To this end the glass is of a special low iron type that allows 90 per cent of the ultraviolet light in sunlight to pass. It is attached by high tension steel cables so that no light is obstructed by traditional glazing bars.
To conserve energy the cooling air is not refrigerated but is cooled by being passed through a labyrinth of pipes buried under the house at a depth where the temperature remains suitable all year round. The house is designed so that the maximum temperature should not exceed 20 °C (68 °F)."
You could always put a pipe under the ground as they did with a fan on the end of it to blow cool air on the plants, but a fan alone will cool the air if it can either blow into a greenhouses or over the plants. People do not realise just how large an air circulation cacti often have in in habitat. When I was up a hill in Chile looking for plants, at one time of the day we were almost blown off out feet by the wind. The old idea of a couple of computer fans in a greenhouse comes nowhere near it, more like a wind tunnel.
Keeping your plants over winter in a greenhouse or a shed with a reasonably large fan, preferably set in the side, with a thermostat set to come on when the temperature gets above a certain limit may be better way of cooling the plants than a fridge.