I have seen a return loop on the hot side, but not returning through the cold side. Do you have more info on this? (I do wonder if recirculating hot through a pipe next to a cold pipe would result in heating the cold)
https://aurorixs.shop/product/watts-500800-instant-water-recirculating-pump-system-with-built-in-timerIn the summer... this is central Texas, so the sun helps keep the water in the pipes hot, so I imagine that the pump is on less often than in the winter. I've not checked though. My gas bills are not out of the ordinary, so I think it's not a ridiculous waste.