But the engine is producing thrust and if necessary, let it burn for a while until one gets enough speed.
You have obviously never seen the blowtorch effect of blazing fuel on vulnerable fuel or hydraulic lines. To "let it burn" is a classic example of head in the sand syndrome. In other words don't look and maybe the problem will soon go away...
If the aircraft is already under control from start of take off run and climb away then any complacent relaxed delay in taking action to extinguish an engine fire warning is nothing less than sheer recklessness.