I will let everyone else tell you why it won't work or otherwise.

HA HA HA

Nobody really knows how to do plumbing. you take that water heater and you can put it almost anywhere you want. now to your showers, you want a separate ¾" line to feed them all the way from the main. That way when someone flushes you don't get burnt. Well it usually isn't all that bad but it isn't pleasant. Still, a properly designed system does not do that.

Because people can't do this now they got valves that regulate the temperature for the shower. And they cost, when all you had to do was to run a separate cold from the main to all the showers.

Even then, to the hot at your sinks, dishwashers, tubs, washing machines, there will still be pipes with cold water in them. And when you stop using it that heat will go from the water into the house.

You want immediate, you said that. They make them. But they are not suited for like when you wash clothes. Made for intermittent use, ten loads of clothes and I bet real money they are history.

People used to take water from the river, run it back to the house and pot by pot heating up on the wood stove filling a large, wide barrel for someone to bathe. (three times a year whether they needed it or not)

They survived. If not we would not have been born.