This ain't a theory, I've done hundreds so its not speculation at all, No it will not cool down anywhere near that fast, but yes it does have to be in use to make hot water, the water enters the tank hotter than your used to, at around 135. The tank becomes storage and as long as you use "some water" you'll be fine. I have a 50 gallon tank and have went to the mountains on a long weekend, return home and still have hot water for a shower 3 days later. I did notice it had cooled down some compared to normal, but I still had to blend a little cold.
Now as far as it heatimg water it doesn't have to?? How's that? The supply line going in the tank provides all the hot water for the home. It is impossible to run it out of hot water, I have installed these in shops where they wash farm equipment as a stand alone unit and they could hook there water hose to it and spray 130 degree water all day.
Like I said I've done hundreds like this and never had an issue, never ran out of hot water, and it's the simplest way of doing it. All winter long and year to year my shower knobs are in the same place, being that it it is very consistent in water temp.
If your boiler is down to 140 and your set up as on demand, well, your gonna notice a big difference, vs as if it were set up where the tank was a buffer, and makes it very consistent