From the original note I am not sure he actually did make an empty threat, he was calling to say that he didn't like the changes and was considering closing. The trouble here is that we all know that 80-90% of the time CC providers do negotiate. I have made very similar calls many times, sometimes with the intent to cancel, sometimes just to see if there was anything they can do to entice or persuade me to stay. 9 times out of 10 in the past (including Amex) they have done something (waived or reduced fees, offered bonus points) and once (also Amex) no go and cancelled.
Have also done similar (successfully) with home loans. I personally would suggest you put concerns in writing rather than just calling up, this ensures it gets in front of the right people, i.e. their retentions unit. Only retentions unit can make deals so getting an "audience" with them is very important, general CSR have very little capacity to move other than just cancel.
But yes, the answer may well be, OK I'll close your account. In that case I'd be saying I want to think about it or asking if they can recommend alternatives (very surprised they didn't do this either). You should only ditch if you want to, sounds to me that OP was actually heading that way anyway but upset to be forced down that track!
