Confusion about Aerodynamic center suppose, i assume Aerodynamic Center lies at Center of gravity.  Then apparently moment by lift will be zero about CoG. But in reality Centre of Pressure may be behind aerodynamic center and still be producing a moment. So isn’t using aerodynamic center absurd?
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.