In contrast to the OP, I had a perfectly pleasant experience with the Extended Warranty. I bought my iPhone last July with my Plat card (subsequently downgraded to the Gold card in January). The phone developed some problems, so I contacted AMEX about it this summer (the Apple warranty had already exipred). They asked me to mail in a copy of the receipt and billing statement. A couple of weeks later, a check for $299 (the original purchase price) arrived in the mail. No hassle (aside from having to wait for the claim to be processed). It made me decide to ensure all of my electronics/big-ticket purchases are made on my AMEX card!
I've heard that the Extended Warranty service is basically rebranded Squaretrade, and it's probably operated by them, too. (That might be why they have limited access to AMEX account information.) Apparently, if you don't have a pattern of filing claims and if it's a smaller-value item, you're more likely to just get a check or credit, though they apparently spot-check claims. Looks like the OP was a victim of this spot-check.
Not sure how to go about filing a complaint about the service the OP received, given that it's a subcontractor. Still, someone at AMEX should be able to hold them accountable for the contract...