Why don’t we use radiators as heat sinks in the summer and pump cold water through them?

Hot water radiators are designed to work with temperature deltas in the 110 degree F range (target 70 room temp, 180 water temp). In the summer your temperature deltas are much tighter, you can only get to at best 32 F before the water freezes and with a target of 70 that's only 38 degrees of temperature delta trying to cool the room. They simply won't work efficiently enough for it to be worth it, not to mention being on the floor is very poor positioning for summer.