Yes, they do. I still have to take time out of my day to audit the cards, challenge the payments, and then cancel the cards and get new ones, and update billpay everywhere else.
No, this generally doesn't involve immediately cancelling your existing card, sending you a new card and canceling the old one when it arrives is a much smoother experience.
Me experience is that at the first evidence of fraud or compromise the card is cancelled. The replacement arrives 7-10 days later. I'm sure it depends on the issuer, but banks in the US tend to a lowest common denominator customer experience.