Light does not behave like a wave some times and like a particle some other times. Light behaves as light. Trying to categorise some behaviour as "wave-like" or "particle-like" is just an attempt to build an intuitive understanding for quantum phenomena by relating them to simpler everyday things like water waves or marbles hitting a wall.
The photodector clicks are not proof of the existence of photons. They are caused by the photoelectric effect, that is bound electrons in the photodetector are in quantised orbits and are only capable of discrete energy jumps. So if anything it's more to do with the quantum nature of matter than with the quantum nature of light. The photoelectric effect even works with
classical (continuum, not quantised) a constant stream of light, not just single photons.
So using the photodetector measurement as a proof of the existence of photons is a bit of an abuse of the photon picture. And it's taking the "photon" picture to the classical extreme of "a billiard ball".
Really, the EM field is a quantum field obeying the wave equation and whose quantum is a photon. Even a single photon obeys the wave equation though, so the better question is:
how to reconcile the (seemingly) localised click of the photodetector with a delocalised photon wavepacket?
A photon is a wavepacket with some spatial extent and a wavefront. It also has a "direction", which we could define as the expectation value of the position operator over time. Hence, there's a spatially varying probability (and hence energy) density. When this gets close to the photodetector, the EM field and the quantum matter interact and cause the photon wavefunction to 'collapse' and to get position-localised (like when you measure the position of an electron in an atom). The position where the photon "localises" is random but follows the probability distribution of the incident photon field, which is a $\propto \sin^2$ and hence different photons cause clicks in different positions on the screen.
To put this in more "usual" quantum terms, then: as long as no measurement is performed, the photon is described by a delocalised wavefunction. When a measurement is performed, the photon localises. Measurements are destructive.
A measurement is performed both by the photodetector and by the antenna. In the antenna case, you'd see a transient discrete signal in one of the antennas, corresponding to where the electron was accelerated by the absorption of the photon.