Elementary particles, and photons are such in the standard model of physics, exhibit, as you correctly say, in the probability density the wave particle duality.
It is instructive to look at experiments of single particles scattering off double slits,
This are single electrons scattering off the two slits, and the wave interference pattern appears in the accumulation, which is the measurement of the probability density for this scattering.
Electrons are massive. Photons display the same behavior, individual scatters seem random , the accumulation has the interference pattern.
The difference is that photons build up the classical electromagnetic wave, whereas there is nothing like that corresponding to the massive particle and probability waves. Zero mass particles can be modeled by quantizing the classical equation. In this case Maxwell's equation, which gives the varying E and B fields is quantized, and the wave function of the photons building up the classical field contain the information of the E and B field. The probability density of any interaction is given by the complex conjugate square of the wave function.
So the superposition of photons ( addition of wavefunctions) can build up in synergy electric and magnetic fields which show the interference pattern in the probability density distribution.
Individual photons themselves when interacting , have only energy, momentum and spin. In an ensemble their wavefunctions are added in superposition thus creating the macroscopically measured electric and magnetic fields of the classical wave. (Any measurement involves the complex conjugate square of the underlying wavefunction of an ensemble).
At the macroscopic level, the mathematical description of varying electric and magnetic fields building up the classical electromagnetic wave, is correct, as it emerges from the underlying quantum mechanical state.( The last link needs the tools of quantum field theory to follow the mathematics).
Thus the classical EM wave is not an outdated concept, it is an absolutely useful mathematical representation, as the technological world we live in shows, in the phase space of large dimensions, and the quantum and classical models blend smoothly at the interface. In the same way that Newton's gravitational model is still valid for the appropriate phase space where general relativity effects are very small.
Yes there is a chain of electric and magnetic fields, it is built up, emerges from the wavefunctions of zillions of photons. See also this answer.
To really understand tha mathematics of what is happening one has to study in depth classical and quantum electromagnetism, and quantum field theory is necessary.