I don't know when the policy changed, but direct bookings were permitted at some point in my memory as well. Regardless of booking channel, it always makes sense to have the rate details confirmed in writing ahead of time.

Amex still has a platform for outside (non-Amex affiliated) travel agencies to book the Fine Hotels rate, although it's no longer supported in the GDS. This platform allows the agent to verify cardmember eligibilty (which the GDS did not). I suspect too many folks with other types of ineligible "platinum" cards were booking directly by phone, resulting in "awkward" check-in experiences for both hotel staff and guests.