Most of my tickets are purchased as E-Tickets from my local City Ticket Office, and because I was there to purchase the ticket in person, no Credit Card is required by me at the airport. On the few occasions when I did buy an E-Fare/E-Ticket over the net or over the phone, I am asked for my Credit Card at the 1K desk at LAX, even though I'm known there. They must enter in the system the fact that the card has been verified, because it is never required again on the rest of the trip.
I've never heard of the credit card policy being waived for Premier Executives or 1K's.
Using Electronic Tickets
When you arrive at the airport, present an official form of identification (driver's license, passport, etc.) to receive your boarding pass. If you use United Reservations, or the Web, you will also be asked to show the credit card you used to purchase your electronic ticket.