The pump causes water to circulate by creating a pressure differential between the inlet and outlet.
The current placement of the recirculation pump is causing the hot water supply circuit to be slightly boosted to higher pressure to push HW through the pipes, and cause it to circulate back to the WH in a loop. Although the intended return path is through the dedicated return line (DRL), water will always take the path of least resistance.
Because the return line is equipped with check valve(s) and possible balancing valves which will act as flow restrictions, any other less restrictive path such as the normally open hot and cold inlet connections on the electronically controlled faucets, will allow the slightly higher pressure hot water to cross over to the lower pressure cold supply, resulting in the cold supply piping often becoming the recirculation return path instead of the more restrictive DRL.
If the pump had been installed directly within the DRL, just before the water returns to the WH, then the pressure differential between the HW supply and DRL would have assisted the check valve(s) to open, thereby allowing reliable flow from the hot water supply into the DRL. The pump in this position will also not create lower pressure in the cold water supply piping.
To further reduce the potential for the pump to interfere with the WH cold water supply, the usual preferred location for the circulation return loop is to enter directly into the bottom of the WH tank, away from the cold water supply piping. When a WH is not pre-equipped with a dedicated return connection, one maybe easily added using a nipple and Tee installed directly behind the tank's drain valve, causing the drain valve to be moved further outward.
I suspect both WH's thermostats are set identically, so with the plumbing in the current series configuration, the tank on the right, will be heating almost all of the hot water consumed, and the left tank's heater will only operate when the right tank can't keep up, thereby allowing cooler water to enter the left tank.
If both WHs had been plumbed in parallel, then each would operate equally to heat 1/2 of all hot water utilized. With both operating at the same time, that will significantly improve HW recovery time, likely close to 1/2 of the current recovery time.
With the tanks plumbed in parallel, then the return loop should be Tee'd to enter the bottom of both tanks.