I agree that a cc user should not have to care about any of this, but people responsible for protecting your money are not so responsible.
(And most people are fairly aware of the fact that card numbers are sensitive information; the newish thing here is a large retailer failing so spectacularly)
The key piece to keep in mind is that the info on that little piece of plastic is never "safe".
If I were warning my mother, I'd just follow your hypothetical above. Of course, if the outcomes of these data breaches start including other identity theft, such as the "$50,000 twitter name"[1] hijack, more realistic phishing attacks, etc., then things are going to get interesting.
[1] Which is of course silly since the TOS specify that they are not to be sold.
They are not easy nor simple AND you can't use your credit card while they mail you a new one
So yeah, as someone who have had their CC misused, it's a pain in the behind