So if you turn the power off to a modern foam insulated tank type water heater and do not use any hot water that water will remain the same temperature indefinitely? If there is no loss it would have to.Did you learn that from your dad? Put your hand on a modern foam-insulated water heater and notice that it isn't warm. Little or no temperature difference between the surface and the air means little or no heat loss.
The tank type might have an energy-bill advantage if the household has heavy users and they end their long showers because the tank runs out of hot water when the otherwise wouldn't.
When smart grids are implemented, large-tank water heaters can be used for consuming renewable energy when it's abundant and storing it for later use.
It is losing heat constantly, unless temp outside would happen to be higher than temp inside then it would gain heat instead. Modern ones just are more efficient.
Bottom line is it takes same amount of energy to heat a fixed amount of water by a certain temp rise. The tankless water heater is just trying to do it all in a very short time and while it is being used. The tank type will do it over longer time at a lower rate, but will also have some losses though they may be minor, while sitting there idle.
That said I still don't think that efficiency difference of electric tankless justifies the need for larger supply equipment or possible demand charges. Gas tankless heaters may be more worthwhile though.