The "leave it running" is coming from old habits, just like "oil change 3000"
Old diesels were hard to start and starters were not long-lived, so with low fuel consumption, leaving it running made sense.
Even last year, I park on semi-truck parking at "Big Texan" and old semitruck pulled over, when driver went inside for order pickup I guess and left the truck running. Darn thing was rattling whole parking lot for about 20 minutes.
New engines don't have those problems and starters become lifetime items.
I even wonder how technology works. Observing rail pressure, it bleeds down to 0, when I turn the engine off, yet the engine seems to start on 1st turn, even after weeks of parking.
That said, having keyless go in my sedan and SUV and dog, I do leave it running on parking, lot, when I go for dinner and temperatures are in 3-digits.
But I always choose the table at the window, where I can observe the car.