Excellent posting Harold as usual EXCEPT the
“A hot water heating system is a sealed system. Water expands about 4% by volume when heated from about 50 to 240F, so an expansion tank accepts this ‘extra’ water.”
Harold ever go into the older buildings where there is no Expansion tank (sealed system)
Only an open tank in the attic with an over flow to the basement or out doors
This of course doesn’t sound like the case I would think the existing tank was disconnected for some strange reason possibly a bad Air Trol tank fitting or the owner had no idea what it was for.
As for the tankless coil I happen to like using a Holby mixing valve exclusively to maintain domestic hot water under a manageable temperature to the fixtures other then the dish washer.
I like 160 deg wash water in residential a dedicated line.