This is from the Amex web site:
Use Your Card
When you are the primary renter, simply use your eligible Card to reserve and pay for your auto rental.
Say "No Thanks"
Decline the collision damage waiver (CDW) or similar option when you are reserving and picking up your car from a Commercial Car Rental Company.
Enjoy the Ride
The Car Rental Loss and Damage Insurance Plan provides coverage up to $75,000 against theft of or damage to most rental vehicles. Coverage applies for the first 30 days, and is in excess to your other sources of insurance.
Note, it doesn't say you must make payment directly to the rental car company, only that you must reserve and pay for your rental with the Amex card.
Hyatt T&Cs explicitly exclude PL from the FFN program. Amex does not.