I don't think it's the correct way to do business to offer a rentention bonus or whatever to customers saying "give me something or I go".

Would you call your favorite airline and say, give me a free first class ticket or I book all my flights elsewhere ?

The correct way is to offer an excellent service and more benefits than other issuers.

The annual fee is not really the problem. For those who don't want to pay it, there are many no-fee Amex cards.
For those, who expect a higher level of service, they can pay for the Platinum or Centurion card fee.