CodgerBill,
Ok, so the "added" pump initially pumps the water that has cooled down in the hot water line, back into the cold water inlet to the hot water tank which in turn circulates the hot water, and replaces the cooled hot water with hot water. One would then turn off the "added" hot water circ pump and open the tap which would now flow hot water.... Right?
But then when one is done washing their face with hot water and wants to take their night-time Aleve with cold water from the cold water tap, wouldn't it be hot, requiring one to run the hot water out of the line before getting cold water? So then it wouldn't it be the same as running cold water while waiting for the hot water... running hot water waiting for the cold water.
It seems to me that a standard hot water recirc line would be a better solution, although, maybe harder,or almost impossible, to install (???).
For those that don't know what a "standard" hot water recirc line is; it's a line at the farthest point from the hot water heater that is run back to the hot water tank so that hot water continually recirculates so the the "hot" water is always hot.
BTW... you didn't show the direction of flow through the pumps, but I think I figured it out.