> Our Mitsubishi has other nice features like it can recirculate the water in the bath through a heat exchanger after you already filled it and got in, so it will keep the water from cooling down to lukewarm
I've also got the ecocute but I was told that it's cheaper to just pump fresh hot water into the bath than recirculate the cooled down water. It uses less energy to pump the already boiled water into the tub than to heat it back up.
I suspect what you were told is a misunderstanding.
I don't see how that is possible. Either way you need the same amount of heat added to the tub. Maybe you save a little because the return pipes are not cooling the water, but otherwise the needed heat should be the same. I personally insulated all my hot water pipes which solves that worry.
Though fresh water and letting out the old will clean the water you are soaking in, which may be valuable.