To the OP--I'll try...
A symmetrical airfoil creates lift in the same way that a non-symmetrical airfoil does. It's (usually) less efficient than a non-symmetrical airfoil, but with the obvious trade-off of being much less *in*-efficient than a non-symmetrical airfoil when generating negative lift (inverted, etc.).
Pressure distribution is a correct way to describe lift, but the old "particle A traveling a greater distance over the upper surface to meet particle B at the trailing edge and thus lowering pressure" is not really accurate or helpful.
A better way to think of it is that a wing creates *circulation* as it moves through the air. That is, the airflow around a wing can be described as the sum of two flows: A translational flow from fore to aft, and a circulating flow moving aft above the wing and forward below the wing (makes sense if you think of upwash in front of the wing, higher velocity above, lower velocity below, and downwash behind the wing, right?). The circulation results in the pressure differential and therefore the lift. It's not necessary even to have an "airfoil" to create circulation--it can also be done with rotation--that's why golf balls slice and baseballs curve. (If you want to see another cool demo for this, take a plain Bic pen, take the cap off the end and pull the ink stick out of it so you have a straight, hollow tube and scuff it lightly with sandpaper. Set it on a smooth table or countertop, and press down hard with a couple of fingers from each hand, then roll your fingers back so that the pen shoots forward with as much forward velocity and backspin as you can get. Watch the pen fly).
Airfoils are carefully designed to achieve this circulation by their shape and angle of attack (rather than by rotation) while creating as little drag as possible. They are most efficient at the speed and angle of attack for which they were designed, but most of them can create some lift even when quite far from their design point--even when inverted. Symmetrical airfoils are just one case of a not-the-most-efficient-possible airfoil--they still create circulation by virtue of angle of attack and the viscous nature of air, the same as a non-symmetrical airfoil.
Does that help at all?
jspeed87, I would respectfully disagree somewhat with your description of stall characteristics. Some symmetrical airfoils *can* have ugly stall characteristics, as you suggested, but other symmetrical airfoils can actually be quite benign in the stall. Much more stall behavior depends on the wing planform and twist, and airfoil thickness and "bluntness" than whether the airfoil is symmetrical or not. Also, a non-symmetrical airfoil will be most efficient for cruise, as it will be able to create the required lift at the lowest drag.
I recommend this description (starting at 4:20).