A lot of the big firms (in the UK at least) will use Corporate AMEX charge cards (rather than credit cards) for this purpose. If they can't pay on their Corporate AMEX they'll often have to go through a more drawn out process to get the payment made by some other method unless they want to make it on a personal card and then claim it back.
* Verified by VISA with my VISA card redirects to a third party domain (not the retailer or my bank) which then asks for DOB and very basic details to set a password (and little more is needed to reset the password). Even if fundamentally secure the user education message of this process is awful. I believe that some card providers make better use and of Verified by VISA mechanisms but not all.
Of course as a merchant it's higher fees, has weird conditions attached to it and virtually all AMEX users have another credit card (indeed Amex provide me with a visa card for when I can't use my amex) so there is little benefit to supporting amex.
Personally, I have an Amex too and I'm not rich or special but maybe it's different in the UK. I use it because I get a ton of airmiles and Amex never seems to get rejected or stopped for fraud checking purposes.