I saw this in action on a smaller scale. In a past job, my wife organized events for a decent sized company. After an event, she'd typically have a $300k+ balance on her corporate Amex. When she went on maternity leave, the person filling in for her job neglected to actually pay the bills, so when she returned there were quite a few emails and voicemails from Amex regarding the over $500k balance.
The messages started as polite and eventually started to get more desperate in tone. At no point were they threatening or adversarial.
I think that this might reflect more on Amex to be honest.
Amex realises that threatening would hurt their business trust more than anything. During the great depression, Amex accepted checks from other banks which were falling and paying through their own wallet as a matter of integrity. Amex has always been built around this idea of trust and prestige.
They make most of money from what I have heard on the transaction fees which are more than others (3% compared to 1%). They might get desperate but I am sure that they are one of the last guys who would wanna threaten you if you are paying some large bills for them (as compared to normal credit card companies which might even hire people to extract your loans in some messy situations)
So perhaps be so rich that the credit card company understands it as well and treats ya differently :-D