You say:
"When I cashed in 4 of the $1000 jetons three weeks ago, the cashier asked for my players card. I'm assuming one of the reasons for wanting the players card was to verify that I had been playing at a level high enough and long enough to have actually won the chips."

This is only part of the reason they asked for your players card.
All transactions over $3,001 are required to be logged for 6A reporting.