If the L/D ratio was less when any flap extension, why could a take-off flaps shorten take off distance (for some aircraft such as the C172)?
looks like you don't have any real piloting experience. never mind, lets make it simple.
First thing: stall speed. in simple words, a speed when there is no longer lift, just a drag. Plane will stop flying. Stall speed is changing with flaps configuration. Stall speed with clean plane (no flaps) is higher than stall speed with full retracted flaps. Which means you can fly slower with flaps extended. And you will be airborne at lower speed (which means shorter take off) with some flaps. (and this is why are flaps 10 recommended for 172 and short/soft field takeoff when no flaps are needed under normal condition) As was already written, you don't care about drag so much during takeoff and you have an engine which is helping you with this. Consider regular GA plane like 172, you will always do a takeoff on full power but you will need something like 60% (or even less) of power for leveled flight (depends on many condition,weight, altitude, temperature,etc) but power requirement will increase when you extend flaps, so you have to add power to keep leveled on same speed. At some point there could be no additional power and you have to choose to stay leveled on lower speed or start descending on same speed.