First, it's not Page in quotation marks. It's just Page, named after Don Page, a quantum gravity researcher. In particular, people talk about the Page time which is when the black hole has evaporated one-half of the initial entropy.
Second, Page's insights are not called Page's insights just by Susskind but they're called so by most of the 200+ followups of Page's (although they may be the only ones who used the new term "Page-scrambled" for scrambling whose entanglement entropy satisfies whatever Page claimed to hold)
Third, Page has outlined arguments that the radiation is very close to thermal, indeed, but because the outgoing Hawking particle is entangled with the dual partner that falls in and modifies the black hole, the whole process of radiation increases the entanglement between the Hawking radiation that is already out and the remaining black hole – or, equivalently, the early Hawking radiation and the late one, for some boundary in between them. The entanglement is very close to the maximal one when one-half of the entropy has already been emitted.
Fourth, the AMPS argument is invalid. See most of the followups
for different explanations why it's wrong. I recommend you Raju-Papadodimas in particular.
Fifth, Preskill-Hayden is right but it is not paradoxical in any way. They show that once the entanglement entropy is forced to revert the rate, in the middle of the evaporation, it becomes possible to extract genuine information about the initial state from the Hawking radiation that is already out, assuming a huge (unrealistic, just in principle, of course) accuracy.