No, you do not. I used this last year after I suffered a concussion on vacation and I was flown from the Bahamas to Miami. Amex handled the triage, booked and paid for all the flights and ground transport. No part of the trip was charged to the Platinum card. The benefit is for the cardholder, period. Was so impressed with this benefit that I made my entire family additional cardholders so when they travel they will be covered as well.
Just curious, do you minding sharing what the cost for that was. The hardest part of getting EVAC coverage is guessing how much is needed.