The problem, as you have described it, sounds typical of customer service at many mass market organizations. I don't think there is any recipe for obtaining satisfaction because your business is not important enough to warrant individual attention at the level required. I know this isn't what you want to hear, I know American Express would hotly deny it, but this is my conclusion. (To be sure there is no misunderstanding, I think this is the case with every mass market enterprise without exception. It's not specific to American Express.)
To add to the above, every organization is going to have a small number of situations that go wrong enough that the cost of fixing them is more than the value of retaining the customer. I think that in such cases, some companies deliberately don't step to bat to fix the problem, even if they tacitly realize it's their fault.