Confusion about Aerodynamic center
It's easy if we clarify your statement a bit.

When aerodynamicists say "moment by lift", this means the derivative of moment by lift. That is, the change of moment per change of lift, $\partial M/\partial L$, or shorthand $M_L$ (or their coefficients more commonly: $C_{M_{C_L}}$). Stability analysis (that's where the aerodynamic center is most useful) is all about derivatives rather than actual forces and moments: we are interested what changes when conditions change.

In this light, "moment by lift" will be zero, true, but "just moment" won't be.