Is the "Radiation Paradox" a problem for a Cyclic Universe?
The explanation does not make sense because a cyclic universe would be completely opaque at the time of the big bounce. It says nothing about the hypothesis that the universe could be cyclic.
In the period close to the bounce, light will be blue shifted by a very large shift $z$. It will freely participate in matter-antimatter creation and the singularity will create and identical initial condition for each cycle in the model.
The same applies to arguments concerning increasing entropy. The second law of thermodynamics applies to isolated systems within the universe. It cannot be applied to the universe as a whole. Entropy is closely related to energy, which is the time component of a vector. Vectors only have local definition in general relativity, there is no way to sum and find the "energy of the universe" or the "entropy of the universe".
Generally speaking, what makes a cosmic cycle a cycle is that it destroys all the characteristics of the previous cycle, such as whether energy is in the form of electromagnetic radiation or something else. So the technicalities of the pseudoscience article are naive and false.
But what about the conclusion? Roger Penrose is famous for developing the twistor theory used extensively in studying particle-particle interactions. He has also developed a cosmological theory of what he calls the conformal cyclic universe (A layman's introduction is provided as his book, Cycles of Time). He tentatively predicted that certain aspects of the previous cycle would in fact show up in the next - not in their original form but as circular distribution patterns of matter and/or energy in the sky. He and colleagues looked for such telltale rings but found none.
Although the preliminary search was negative, this does not disprove the whole conformal cyclic theory. It remains an open possibility that the Universe might be cyclic and, further, that something identifiable could leak through from the previous cycle. However the idea that this something will survive multiple cycles, so that we could count them like tree rings, has no foundation whatsoever. So the pseudoscientist's conclusion is also unsound.