Don't worry about the 1/2. Coefficients in an equation are there just to give you a number, not really to give you understanding. The 1/2 could just as easily be embedded into the Cl, although then some other equations would need a factor of 2 included.
The funny thing is that the 1/2 is typically kept in the equation to aid the understanding. As has been mentioned before, that number just falls out of the mathematics and not the physics. And here we are concerned with the physics, but wondering about the 1/2. Be much more concerned with the exponent on the airspeed term...it actually matters.
As far as having a different understanding than Bernoulli, don't be too concerned. These theories do not define how nature behaves, they are just an attempt at quantifying how nature behaves. You can describe many systems using different physical concepts without revealing an inconsistency. An example of this is in basic kinematics, describing the motion of an unaccelerated particle. You can give its mass and momentum, its velocity and momentum, its velocity and energy, its momentum and energy, etc. Just because you're using different terms and concepts, doesn't mean there is anything different about nature.
Bernoulli's theorem is consistent with everything known about classical mechanics (for fluid dynamics, perhaps not for airfoils). You can define static and dynamic pressure, and then demonstrate that static pressure in a moving fluid decreases with the velocity of the flow. You can then demonstrate it experimentally. If you want to describe it using concepts other than static and dynamic pressure, go ahead. Doesn't mean that Bernoulli was wrong.