I have done many batches using the "heat exchange recirculating mash system" you describe. Temperature control of the hot water does require fairly constant attention, which is a bit of a pain. Also, a large volume of hot water makes temp changes go pretty slowly. But with a very low, constant flame it has not been necessary to stir the water because the convection in the kettle will keep the water moving around the coils.
Also I have had good success with adding a thermowell and Ranco digital controller to the mash tun (10 gallon round Rubbermaid cooler, thermowell w/stopper goes in through a hole in the lid). The controller shuts down the pump at the set temp deep down inside the mash. For the next step, just dial in the new temp and the pump starts up again. I try to keep the hot water to within 8-10° Of the desired mash temp, for fear of denaturing the enzymes on their trip through the coil. I think that's probably just being paranoid, tho.