I think you are assuming a continuous slope in the aerofoil-polar, which isn't usually true, there are many aircraft where past a point, the increase in Cl.max with increasing flap deflection becomes negligible, yet the increase in Cd0 becomes quite large.
Not sure that's the assumption. I think LOMCEVAK is just assuming a monotonic decrease in L/D as flap is extended.
Thus, whilst L : D is improved to an intermediate flap setting, it is worsened again as further flap is selected - giving a poorer take-off run and climb gradient in most cases. Not a bad thing, we all rather rely upon large flap settings for landing.
While I don't think it's possible to generalise across every possible aerofoil and flap combination, but I think many if not most have best L/D with flaps retracted.
This is the reason that many aircraft (including the afforementioned C172 I think, and certainly a PA28) take-off with an intermediate flap setting, and on a go-around the immediate action (following full power selction) is to raise the flaps to that intermediate setting.
If that were really best L/D, wouldn't it be recommended for best glide and best rate of climb (well the latter is a little more complicated but tends to be around best glide speed)? I'm sure you're right that the L/D improves significantly as the flaps are retracted to intermediate, but it doesn't mean that it doesn't improve still further when the aircraft is clean. They take-off using flap to minimise ground roll, not to maximise climb performance.