I saw this mentioned somewhere else too. Really odd. Presumably they’re only making this “offer” to cardholders whose credit profiles suggest trouble in the near future, but the ones who can actually come up with the cash to pay off their balances would certainly be very disproportionately from the subset who would NOT end up having trouble paying. So AmEx would end up blowing $300 a head to get rid of good customers, and be left with all the ones who have no way to pay off their balance even to get a $300 cash gift.
People who get the letter and don’t pay off the balance lose their cards anyway - and AmEx will be writing off the balances. I presume if these were good clients to begin with, they would not be getting the letter. What they are attempting here is to rise to the top of the priority list of debtors, a rather moot point. If people could pay up, they would.