Here is what I think is a more modern understanding of the situation. It is true even in classical physics that a circularly polarized electromagnetic field carries angular momentum, and that E = L*omega, where E is the energy, L is the angular momentum, and omega=2*pi*f is the angular frequency of the EM field. See Feynman I-33 last paragraph, https://www.feynmanlectures.caltech.edu/I_33.html
Now, the experimental fact is that L is quantized in multiples of hbar=h/(2*pi). Thus you can derive Planck's results from the assumption that angular momentum is quantized. So the photon is really a quantum of angular momentum, which happens to have a non-quantized energy. But Planck's didn't know that, and worked with quanta of energy proportional to the frequency. (It's not like Planck was wrong. His theory was a necessary step towards the discovery of the deeper result that angular momentum is quantized.) Planck's constant of proportionality has the units of "action", which is the same units as angular momentum but "means" something different. The author starts with energy=action/time and attempts to reverse engineer the energy into quanta of energy, which as far as anybody knows do not exist, instead of starting with energy=angular_momentum*angular_frequency, where the angular momentum is truly quantized.
The author's theory falls apart when you realize that angular momentum in general is really quantized in units of hbar/2 (not for photons but for electrons), in which case her theory of energy being quantized in terms of a full oscillation of light becomes untenable, since you would have to quantize in terms of half an oscillation, at which point why 1/2 and not 1/3 or 1/4? Whereas there are legitimate geometric reasons why an object rotated by 360 degrees would change sign but yield the same physics. See https://www.feynmanlectures.caltech.edu/III_06.html