If I had to guess, the most problematic point would be consistent large payments from the supplementary cardholder's bank account. Since I never had any AUs -- I am kind of surprised they allow transfers from anyone apart from the principal cardholder.

Since the primary account holder will be liable for all charges and AU has been paying instead. There is little record of the main account holder's ability to pay. The AU can decide to put a bunch of charges on the card and then disappear without a trace. Amex (or any other lenders) may see this as a high risk.