So what I misunderstood was that the "normal" concept of a wave function doesn't apply but "A few substitute wave functions have been suggested ..."

May I suggest as the wave function, the classical electromagnetic field equation that would describe the light at high intensity passing through this region, times some very small constant. After all, such an equation describes, in the case of full-strength radiation, the intensity we would observe if we took intensity measurements at various points, and thus corresponds, in the case of very weak radiation, to the probability that we would detect a photon at any particular place and moment of time if we tried.