When you quantize Maxwell’s equations to get QED, the equations still hold. All that changes is the meaning of the potentials and fields; they are now quantum-mechanical operators. The field of a charge still extends arbitrarily far from the charge. In fact, the electromagnetic field exists even when there are no charges. For example, it has “quantum fluctuations” in the vacuum state. And a Big Bang does not require charges to produce photons; expanding spacetime is sufficient to produce them. Your conception of photons as *being* the field is incorrect. They are the quanta of the field. The quanta interact with matter as point particles, but the quantum field exists everywhere, just like classical field in classical electromagnetism. Finally, each charge does not have its own field. There is only only one quantum EM field pervading the universe, and all photons are quanta of this one field.