I think in the past one of the issues has been duplication and alteration of license copies. That was behind the requirement of having a signed in ink, in a color other than black, original copy of a license mailed to the other FFL holder before shipping. With modern copiers and photoshop programs, even with changes to the appearance of the license about six-eight years ago, it has been too easy to alter a license and receive shipments to a non-licensee.
The change recently has been the availability of the EZ-Check system where information on a license can be verified from the ATF website. Since this has been available, I have even checked the original copies to make sure they are correct. With this ability to double check, accepting a faxed copy is less of a risk.
One consideration is that, given just a fax number by a customer, how do I know where the fax is going? If I mail to another FFL holder, I know it's going to a legitimate address. I don't know who or where that fax number might be. I could do some checking, but if I have to take a half hour to go around the horn to make sure everything is on the up and up, then I probably can't do $20 transfers, there is just too much of a time investment. For a regular customer, there may be a return on the investment down the road and everyone will win. In a situation as first stated, where the regular dealer isn't available and someone is insistent that the process be worked through for a one time transfer, it's not worth the time investment.
Call me paranoid, but I'm always hesitant when someone insists I do something outside the normal routine, and wants it done immediately.