All that happpens when you run out of fuel is that the engines stop.
Are you using some of that famous "British Understatement" when you say that?
When either the last engine or all of the engines quits....a lot of things happen in addition to it getting awfully quiet.
Generators cease providing power, lots of electrical powered systems stop working, and the Pilot's workload becomes extremely intense. Throw in flying over a big city....and darkness.....and it gets complicated for sure.
You are factually correct to the extent when the engine(s) stop getting fuel....they cease operating.