There's no "idle thrust" on Falcon 9 since even the lowest thrust is enough to lift the almost empty booster. You also don't want to completely run out of fuel because your engine will probably explode. This was just a residual fire.
Can anyone suggest keywords/links about this? Sounds interesting.
The fuel is also used as a coolant [2] so you would not want to run out of fuel in the plumbing before you run out of propellants in the combustion chamber.
[1] https://en.wikipedia.org/wiki/Turbopump
[2] https://en.wikipedia.org/wiki/Merlin_(rocket_engine_family) (search "regeneratively cooled")