Thank you LegionOnomaMoi,
In QED "all-path" argument
I'm not entirely certain what you mean to refer to here. If by "'all-path' argument" you mean the sum-over-all-paths approach to quantum theory better known as the (Feynman) path integral approach, then this is neither an argument nor relevant. Photon numbers are not in general well-defined in any theory, but in QED as in relativistic quantum field theory in general particle number is not only ill-defined generally it is not a conserved quantity.
we do talk about a single photon
Often we do talk about single photons in various fields both within physics and in other sciences. Likewise, we speak of electromagnetic waves, of Newtonian gravity, of magnetic fields, and other things that are merely terms referring to useful concepts in theories we know either to be wrong or at best to be useful approximations. But when we speak of e.g., single-photons in e.g., reference to quantum control or quantum engineering or any number of other technologies which by their design detect single photons it is important to be aware of the fact that we do so not because we believe we have somehow made photon number a well-defined concept or produced experimental evidence against relativistic quantum theory. Rather, it is because the tools we use to prepare, measure, and manipulate physical systems are limited by their design. We can build devices to send and receive signals that we call radio waves and which we interpret and analyze using classical electromagnetism, just as we can analyze the collective behavior of molecules making up the ocean by speaking of water waves and acting as if these were indivisible, nonlocalized waves.
When, however, we wish to speak in terms of what our best evidence and theories say about the world around us, and not in terms of useful approximations, then we must be more careful about how we speak. AM radio waves don't exist as more than at best descriptions of phenomena that are useful but break down at a particular level. Just as it can be useful to speak ocean waves rather than their molecular (or atomic) constituents, so to it is useful to speak of electromagnetic waves or even radio waves. But we know that ocean waves are composed of various molecules and we know that radio waves and electromagnetic radiation more generally are course descriptions of a very different underlying reality that can't be described by E & M.
Your statement sounds when QED gets into trouble describing a phenomenon
It is arguably the best theory in existence and the measurements obtained by QED are more accurate by far than those in any other field. It is certainly not troubled by perceived contradictions with AM radio waves.
You seem also to be conflating quantum mechanics with QED. It is true that in quantum mechanics and when using certain devices for which classical E&M is not adequate it is convenient to refer to single photons which can be treated to a fair approximation in QM in which particle number at least is conserved (or in non-relativistic QFT and a semiclassical approach to electrodynamics).
But the kind of talk one finds in textbooks, monographs, and so forth on quantum mechanics (let alone in popular accounts) concerning photons is usually reserved mostly to introduce the failures of classical theories and in particular Einstein's 1905 paper that described quanta of light.
In reality, the descriptions of photons one encounters in texts on quantum mechanics (or in popular literature) are necessarily incomplete and known to be inadequate. Maxwell's equations concern fields, and quantum mechanics doesn't. Quantum mechanical descriptions of the dynamics of systems are not Lorentz invariant, and whatever else light is or is not, it is most certainly a relativistic phenomena. Quantum mechanical states are given by rays in Hilbert space even in non-relativistic QFT one must content with the infinite degrees of freedom required for the quantization of electromagnetic fields (whether one uses the Dirac equation and Fock space or path integrals or whatever).
So-called "second quantization" or the extension of quantum theory to interactions (including e.g., the interaction of a photon with itself or an electron with its field) involves considerable difficulties that took many years and many great minds to overcome (or even to understand) and even graduate level texts on quantum mechanics will not generally include more than a hint of the problems QED solves such as the nature of light. Precisely because QED developed after we knew any fundamental physical theory must describe systems in a manner that incorporated time on an equivalent footing with space, nobody put that much work into trying to formulate a theory of electromagnetic radiation which relied on the classical phase spaces of point-particles. Actually, the original Schrödinger equation was a relativistic wave equation now known as the Klein-Gordan equation but Schrödinger rejected it because it yielded solutions which were untenable. But even the Dirac equation had similar issues. It wasn't until we had a better empirical basis for understanding "negative energy" states in and backward-timelike trajectories in terms of antiparticles like the positron (not to mention the extremely helpful Feynman diagrams, which enabled complicated mathematical terms to be pictorially depicted and analyzed) that we were better equipped to understand what the mathematical content of such equations implied physically.
Finally, just as for many purposes one can use classical E&M to develop, design, and utilize a vast array of sophisticated technology, one can go beyond classical field theory without having to deal either with the more sophisticated tools of relativistic kinematics or (worse yet) relativistic QFT. But again, in doing so one must recognize that such uses are approximations.
And in this case dismissing the existence of individual particle. So, either those Experimentalists are falsely boasting the capability of their equipment, or the theorists are failing to produce a comprehensive theory that clearly explains those phenomenon. Science cannot have it both ways!
Experimentalists work with devices and experimental designs that are informed by theory and which yield results interpreted by theory. One speaks of electromagnetic waves or single-photon detection in experiments not because one believes experiments require or even support the existence to such phenomena as such. You can buy a magnet and use it to make certain things move without touching them, and we can speak of this in terms of magnetic attraction and magnetic fields, but this doesn't suddenly undo the work of 19th century physicists whose work culminated in the unification of electricity and magnetism into a single physical theory.
In fact, the word "particle" in modern physics is something of an unfortunate tradition. Modern particle physics deals with unobservable interactions with unobservable and undetectable systems using a convenient, fictitious language and a correspondingly precise, rigorous mathematics.
Particles like photons are terms used to describe groupings of patterns of results yielded by experiments/detectors appropriate to high-energy subatomic processes. Theory is required to make any sense out of the chaotic results of high-energy collisions or even tracks in cloud chambers. QED is such a theory, as is the standard model of particle physics which incorporates it. In this as in all such models, quantum mechanical descriptions require taking into account energy fluctuations and therefore particle creation and annihilation. The dynamics of such systems, when described relativistically, are given by equations in which single particles can't exist (the vacuum state is a many-body system). Already in quantum mechanics the problems with treating systems as isolated as in classical physics may be readily seen in the issues of nonseparability. But in relativistic quantum theory, one has to contend with the dissolution of even the fiction of single particles in principle.
