Which one is more fundamental in nature: matter or radiation?
He basically shows that a local symmetry of the matter field described by Dirac equation will directly give as a consequence the existence of the gauge boson field, here photon.
This is wrong. There are perfectly well-defined theories where you have matter fields but no gauge fields. And vice-versa: we have perfectly well-defined theories of gauge fields with no matter. These two objects are conceptually independent, and the presence of one does not imply the existence of the other.
The correct statement is that the presence of matter fields provide a natural motivation for gauge fields. But the opposite statement is also true: the presence of gauge fields provides a natural motivation for matter fields, so both these objects are at the same conceptual level: neither of them implies the other, and they are both good enough in order to motivate the other. Form a logical point of view, neither of these fields is more fundamental than the other.
Can we say that radiation has evolved from matter? Does this contradict our cosmological models that say the early universe has been radiation dominated, now matter dominated, and in the future vacuum dominated?
Apple and oranges. The fact that the energy density of massless particles, of massive particles, and of the cosmological constants evolve through different powers of the red-shift parameter $z$ has nothing to do with one being more fundamental than the rest.
''He basically shows that a local symmetry of the matter field described by Dirac equation will directly give as a consequence the existence of the gauge boson field, here photon.''
But it is the local symmetry, not the matter field, that is responsible for the existence of the gauge boson field. Indeed, the local symmetries of a quantum field theory are in a 1-1 correspondence with the gauge boson fields.
Thus it is the assumption of local symmetry that forces the photon, not the presence of matter. There are lots of (perturbatively defined, nonrenormalizable) relativistic quantum field theories involving matter but no radiation.
In nature, both matter and radiation are fundamental.
This is a Chicken and Egg problem.
Geometrically, it is more natural to say that Gauge bosons (radiation) are consequence of having Gauge symmetry in fermions (matter). Gauge bosons play a role of connection in a vector bundle formed by fermions and their transformation properties. In fact, the boson kinetic term in Gauge theories is none less than the product of the curvature tensors!
More over, there is a simple reason for having Gauge-invariant matter: the local phase in QM must have no effect on the measurements since it disappears from the probability. This is another explanation of starting from the fermions and getting bosons from the gauge symmetry.
QFT has very profound geometrical aspects. Although, it is a quite a big detour, I'd recommend taking a look at some Differential Geometry textbooks. My favorite is "The Geometry of Physics: An Introduction" by T. Frankel.