Why does the moment coefficient for angle of attack = 0° not have to be negative?
An airplane achieves steady flight at a given airspeed only if the total pitching moment about its CG is zero. This is achieved with a particular fixed tail angle/elevator angle/canard angle. If the AOA corresponding to the total lift at this airspeed is 0 AOA, then so be it. It's just that for typical airplanes at operational speeds (away from $V_{FE}$ for example), the AOA is typically larger than zero; this means that, for a typical plane, the pitching moment at 0 AOA must be larger than zero at the tail angle, elevator angle, etc. corresponding to the trim speed for static stability.

Why larger than zero? Static stability means that when perturbed from the trim AOA (at which the pitching moment is zero), the resulting pitching moment will return the aircraft back to its original trim without pilot compensation. Take a look at the graph below; note that the moment is zero at AOA=1.5 (i.e. the trim condition). 



If the pitching moment at zero AOA is negative (the orange line), then the airplane will diverge from its trim point when perturbed. That's the opposite of static stability.