2 Things...

Would paying for First Class count as a form of payment for hanging the coat? Even a "free" upgrade is essentially paid for with Medallion Status. Going back to the valet scenario, are you saying if I use a complimentary valet they are not responsible just because it is free and optional??

Delta took possession of the coat and they are responsible for it. If I handed my carryon to be gate checked and it disappeared they would be responsible. If the FA lifted my carryon into the overhead near me I would say she did not take possession of the coast simply helped me store it. If she had taken the carryon and stored it somewhere else that customers are not suppose to access (cargo hold, closet) then yes, I would expect Delta to accept responsibility.