What is the motivation for assuming "Page" scrambling for Hawking radiation?

Obviously, at the semiclassical level, we want the outgoing Hawking radiation to look thermal and mixed. However, surely there are possible pure states which are scrambled enough that it looks effectively thermal when not nearly all of them are taken into account, but yet not satisfy the "Page" property. Only Susskind calls it Page, but whatever...

That if you take less than half the Hawking radiation, it's maximally entangled with the rest. Is this assumption far stronger than necessary?

It leads to bizarre conclusions like Presskill-Hayden and AMPS.