Classic FT...one person misreads and the rest gang on.
OP said the phone agent said 'normal protocol' is to get vouchers.... but THEY DID NOT GET THE HOTEL VOUCHERS.
it isnt 'unclear' one of them is lying. If the person at the counter is challenged by language I would not be surpirsed to hear AA is not being forthright.
As you are EXP, I would ask an EXP agent to document the PNR. Then ask them for the line number in the PNR. Give this to the gate agent and say "line XXX in the PNR state this was a crew delay and we are due accommodations". Find a supervisor if necessary.