Why are Flight Directors turned on so early?
For a jetliner, the 'why early' is to test the auto flight system. Turning on the F/D also turns on the Flight Mode Annunciator (FMA) and displays the armed modes.

So it's an early check that all systems are go.

From a 777 manual, any fault will be indicated on the FMA as it reads above.

Here are the flow checks pertaining to the F/D and FMA during the cockpit setup: