It is often said that quantum effects only become manifest in loops, and all tree-level calculations are classical. I am trying to figure out to what extent this claim is true. I know the claim arises from the expansion of the quantum effective action as a power series in $\hbar$, with the leading term being the classical action. Another way to see this is by counting the powers of $\hbar$ in a Feynman graph and noting that there is one power per loop. This is all fine.
However, when it comes to calculating an actual, tangible cross section at the classical level, I believe the claim breaks down. Take, for instance, pair production ($\gamma\gamma\rightarrow e^+e^-$), a process that has a tree-level contribution in the quantum theory. As far as I can tell, this process cannot occur at all in the classical theory. Consider the QED equation of motion for the fermions, $$(i\gamma^\mu\partial_\mu-m)\psi=e\gamma^\mu A_\mu\psi.$$
Classical scattering can be thought of as the (deterministic) interaction between two wavepackets. If the process is $\gamma\gamma\rightarrow e^+e^-$, then initially we must have $\psi=0$ and the field $A_\mu$ will be the sum of two localized wavepackets which will move toward each other and eventually overlap. However, we see from the above equation that if $\psi=0$ initially, then it must remain zero at later times, since the source term is proportional to $\psi$ itself. In other words, there is no way to excite the $\psi$ field.
Is the above argument correct? Is pair production impossible in classical field theory? Is the claim that quantum effects are in loop corrections just sloppy, handwavy nonsense?
Note: I know there are related questions here, here, here and here, but the answers do not go beyond what I mention in the first paragraph.
