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Firstly, FRs aren't all that common, but what seems to annoy people the most is:
1) Sudden suspension of all the cards, followed by a demand for the documents in a short period of time. Sometimes, the card member is out of the country and that can make things hard.
2) As LS said, intrusiveness. This is particularly the case when you file jointly and have dependents, but the card is your own. Suddenly Amex is given a lot of info about people who have no role in the card in question. People feel that pay stubs and bank statements should suffice, although these are much less secure from Amex's viewpoint.
And while people are accustomed to this for "big" things like mortgages, other credit card companies don't do this. As Amex defenders point out though, if the other companies have a concern, they simply close the card. At least with an FR, you have the option to attempt to defend yourself, and certainly some people keep their cards more or less intact.
But, while I supplied the transcript for a CLI (something I request) , I wouldn't do it for a FR now!