chaadster

First, I gotta say the "fixed gear are quicker" thing is absurd, and I will eat my derailleur if I'm ever beat across an intersection by fixie. Can't happen, won't happen...and by the time I'm 3/4 across spinnin' 95rpm they're just hitting 85rpm while I'm snatching the next gear and am gone...

To the OP, I don't see how this foot push-off is working, in that your lead foot should be on the pedal and at the top of the power stroke, i.e. 1 o'clock. When the light goes green, you should be jumping on that lead foot and driving forward with your full weight. Your down foot, or trailing leg, has no leverage to push; if it does, you're doing something that is far less effective at propelling you forward.

Furthermore:

Have your hands in the power position, on the drops like Caloso talked about, or out on the barends if you've got those on your flatbar (if you don't, get some).

Lift off the saddle on launch, putting all your weight and force into driving that pedal through the lower stroke.

Practice getting your down foot on the pedal as quickly as possible, so that it can be there to work when that pedal hits the top of its stroke. This is key: seAmless, quick, power.

Are you using foot retention? You outta be. It's more secure and helps you deliver more power. However, don't worry about getting the trailing foot in on that first stroke, or even before clearing the intersection; you need to focus on pedal speed and confinuous power. Once you're clear, flip the toeclip or get the cleat in then; as you improve your coordination, you'll get proficient and getting your down foot secured immediately, but it's nothing to worry about up front.

Launch in the right gear that allows you to accellerate to your max cadence by just past midway of your sprint zone, and get on that next gear and spin it back up ASAP.

That'll get you goin' good.