This doesn't warm up the water to the heater, but rather the water going to the cold side of the mixing valve in the shower.
If it takes me 5 minutes to get the stupid thing at the right temperature now, imagine what will happen when the temperature of the incoming cold water depends on the temperature of the drain water - and it's a positive feedback - the hotter the water, the hotter the incoming cold water, making the output even hotter, etc.
This has the potential for using more water rather than less. Although I assume not all faucets are as sensitive as mine - in less than a degree or so of turning I go from freezing to boiling.
According to their actual website, you can set it up so that it feeds the warmed water back into the water heater, rather than straight back into the shower. Apparently there are thermostatic mixing valves that would probably help with your problem as well (although I just read that in their FAQ and really don't actually know what that even means).
Of course you can feed it that way, but the plumbing would be very convoluted - up to the shower, through the machine, and back down to the heater. The loss in water pressure would be bad, and can you imagine having two showers? The plumbing would be nuts.
I know about thermostatic mixing valves, but I'd have to break the wall of the shower, and redo all the tile. But I guess if I'm putting in this device I could do it at the same time.
But all the thermostatic mixing values I know of actually work on water pressure - so if someone turns on a faucet it automatically adjusts to keep the temperature correct.
I don't know of any that actually measure the temperature of the water, which is what is needed to help this device.
If you want to do this properly you install a heat exchanger on your main sewer stack, and then route the incoming water line through it. They do exist, this concept is not new.
Thermostatic mixing valves use a thermostat to control the ratio of hot and cold water to output a selected temperature. Basically you select a desired temperature and the system attempts to provide you with it. This let's you use the shower while the hot water pipes are still heating up even though the hot water's temperature is changing.
If it takes me 5 minutes to get the stupid thing at the right temperature now, imagine what will happen when the temperature of the incoming cold water depends on the temperature of the drain water - and it's a positive feedback - the hotter the water, the hotter the incoming cold water, making the output even hotter, etc.
This has the potential for using more water rather than less. Although I assume not all faucets are as sensitive as mine - in less than a degree or so of turning I go from freezing to boiling.