I think OP is talk specifically about hot water radiators.
He is asking why those rads specifically can’t have cold water piped through them in the summer.
I think the answer for that is: they weren’t designed to manage the condensation that would occur if the radiator (or pipe) temperature is lower than the dew point. Also, hot water can be a lot more above room temperature than cold water can be below. I think a lot of radiators are actually supplied with steam, not hot water which also lets them potentially use the latent heat of condensation of the steam for even more heat transfer.
I think OP is talk specifically about hot water radiators.
He is asking why those rads specifically can’t have cold water piped through them in the summer.
I think the answer for that is: they weren’t designed to manage the condensation that would occur if the radiator (or pipe) temperature is lower than the dew point. Also, hot water can be a lot more above room temperature than cold water can be below. I think a lot of radiators are actually supplied with steam, not hot water which also lets them potentially use the latent heat of condensation of the steam for even more heat transfer.