Camber directly affects the amount of circulation an airfoil can generate, adding camber will always generate a higher zero alpha lift coefficient (intersection of the lift coefficient curve with the x=0 axis) up until a point that there is too much camber to sustain attachment.
This leads directly to the question asked - does camber increase CL_max? For any given AOA, there is an ideal suction side geometry that generates a pressure distribution that is always on the edge of separating. Liebeck defined this ideal shape but only for laminar flow, for which the boundary layer equations are closed form, this was all based on the work of Stratford who came up with the ideal pressure recovery. Having a surface that is about to seperate everywhere can even be true for a symmetrical section at some inclination, although it's not going to look like a NACA 0012/15 or anything familiar. The effect of the pressure surface on the amount of circulation generated, is much more insignificant than the suction surface, but it is not zero. Therefore, a cambered airfoil will make slightly more load at max CL than a symmetric section.
In reality, there is no magic button to increase, decrease camber. Airfoils are complex geometries that cannot be driven by one parameter without affecting everything else. Adding camber by 'bending' the airfoil somewhere near the front of the chord will give some more circulation, but also probably increase the departure angle, and create a risk of separation towards the trailing edge where the pressure recovery is too steep for the boundary layer to cope with.
Unfortunately there is no easy +camber = +max lift coefficient relationship, but for a cambered and non-cambered airfoil designed to operate on that separation limit, the cambered airfoil will generate slightly higher max lift.