The airfoil sections for all modern wings and rotors are not symmetrical, since they are optimized for lifting, which occurs one way. The shapes are selected for their ability to more efficiently develop lift wthout requiring more power.
Moment is a very secondary reason for selection of the airfoil, because even very poor moment characteristics can be solved with more structure (weight) in the swashplate and controls/servos. To double the weight of the controls might add weight equivilent to perhaps 1% of the rotor thrust, while a very optimal airfoil might add 3% to the net lift, so why not push for efficiency of lift?
We speak of figure of merit, which is the efficiency of the rotor as compared with a perfect one. Where all the engine power is converted to pure lift, we say the figure of merit is 1.0. Older symmetrical airfoils and simple tips produce rotors with figure of merit of perhaps .70 to .72. Modern assymetrical airfoils and swept tips produce figure of merit in the .78 to .80 range.
That means a 10,000 lb helo with an old rotor system could become an 11,400 lb helo with the same engines, at least as far as power efficiency is concerned.
There are some ways to see this in the flight manual, using the "cruise charts" and climb charts in the performance chapter. If ppruners are interested, I can try to post some examples.
In any case, the use of CFD (computational fluid dynamics) has produced a new set of extremely efficient rotors with carefully designed airfoils that move air and create lift very efficiently. These airfols are not symmetrical, by design.
It is my experience that vibration has no relation to symmetry in the airfoil. Why should it? What has symmetry about the chord axis to do with anything "balanced" about the way the rotor moves air? Most blade induced vibrations are due to the structural tuning of the blade itself (which produces harmonic forces at the hub) and the interaction of the blade with the shed vortex from the previous blade in the sequence. We often tune the harmonics with weights along the spar to try to make the blade spin more harmoniously.