I doubt that idea is to really cool off, more of balancing out temperature.
When I drove Volvo 240 as daily, that was really important as 6 cylinder engine other end did see completely different temperatures than other end, front of engine was seeing wind and rear hardly at all, so if that was switched off immediately after drive you got warped and cracked head, really many of those motors had that destiny.
Nothing to do really with cooling off.
Here no vehicle is allowed to idle more than 2minutes, unless it is really cold or if work is being done with accessories that require motor to be running, but nobody really follows this practice, nobody is controlling that law, so it is not rare to see diesel car idling at front of shop door for 15-20 minutes, blocking two handicapped parking spots and the doorway.
__________________
|