In your very question, when you describe the Dirac equation, you automatically imply the answer that the most fundamental concept in nature is mathematics. It is not. Mathematical models can be very good, but mathematics is simply logic limited by the initial assumptions and the nature always goes beyond any model. So if your question is limited to some particular model like the Dirac equation or QFT, then your question isn't truly about the nature and you already have several answers like that. However, if your question is about the deep properties of the nature beyond any specific mathematical logic, then you must use physical intuition. It is not guaranteed to be always correct, but it is the only instrument of kniwledge that goes beyond the limitations of mathematics.
Another limitation that you must overcome to get a proper answer is the limitation of the "conventional science". Not every theory accepted by the "consensus" is always correct. One example from history is, "the Earth is flat". People were burned alive for trying to prove otherwise. The "consensus" is a very powerful limitation.
With all this out of the way things may suddenly become more clear. By "radiation" you seem to mean specifically the electromagnetic radiation. Although other interactions may follow the same trend. The electromagnetic radiation is emitted and absorbed by electrically charged particles. They can exist without radiation, but radiation cannot exist without them (see more on this below). So the answer in the scope of your question is that matter is more fundamental than radiation and it was matter that produced radiation in the first place. In fact, it is the same conclusion that you came to by analyzing the Dirac equation, so this conclusion does not even need to go beyond this particular mathematical model. Does this answer follow the current cosmological model? Well, It doesn't really matter. You cannot comply with every "accepted" theory while asking a fundamental question. Plus the current cosmological model is work in progress that has problems much bigger than this one. So let me summarize.
are we saying that the theory of electrons predicts the existence of photon?
Absolutely. If it didn't, we'd need a theory that does, because photons are emitted and absorbed only by charged particles.
In other words, which is more fundamental in nature? the matter or radiation?
Matter. Although the question is a bit moot, because both are the integral parts of existence, which ultimately would not be possible without both.
Can we say that radiation has evolved from matter?
Yes. Electromagnetic radiation is emitted only by charged particles. The opposite would only be possible via the two photon physics that has not been observed and brings more problems than solutions. For example, photons don't experience time and cannot decay without hitting a real charged particle (same issue as that the neutrino oscillations are impossible with massless neutrinos). Until actually observed, the two-photon interaction is unproven. Another concern is that the matter-antimatter disbalabce is much easier to reconcile conceptually at the moment of creation than later on.
Does this contradict our cosmological models that say the early universe has been radiation dominated
Not necessarily. Matter of the very early universe could have emitted enough radiation for the early universe to be radiation dominated. In either case, the current cosmological model has much bigger problems to worry about and is hardly a high standard for fundamental questions.
Finally, is this answer absolutely correct? No, of course not. Like anything else it is limited by our knowledge and understanding that constantly evolves.