(1) The savings for "on demand" isn't in the water for the shower, but the water that cooled sitting in the tank and was then reheated. (2) Many water heaters in the US are natural gas (or propane), not electric. It's far more efficient to burn NG to heat water than it is to burn it to make electricity and then use the electricity to heat water. Given adequate insulation (so water isn't reheated), a stored-NG system can easily use less energy than an electric on-demand system (3) "on demand" water heaters can increase peak use. Stored hot water can be heated off-peak.

I don't know if there are on-demand water heaters that use NG.

There are and they are quite popular here in Alaska, where the "cooling off" of heated water is obviously quite a challenge. Problem is they are not widely adopted because of the very high initial cost of purchase. They do eventually pay for themselves, but it's a save now vs. save later problem.

Why do you have a hot water heater? Personally I have a cold water heater. :)

Few people have on demand water heaters because the power is too great for most wires (even 400amp house can maybe manage 1.5 utilities if you live in a northern climate where the incoming water is near freezing), and using gas doesn't work quickly enough. Just insulate the tank really well.

Using electricity to heat water is very wasteful. But I guess if you don't have gas you don't have a choice.

Using gas to heat water in instant/tankless fashion most certainly works just fine. There are off-the-shelf units that can deliver over 5 gallons per minute at 75*F temperature rise.