There are ways, in an apology, to focus on feelings such that the apology is ineffective or even insincere.
For instance, suppose I call someone stupid, and that person feels bad. Someone else suggests that I apologize, and so I say: "I am deeply sorry that you feel bad because I called you stupid". Notice how this is not really an apology at all; it only begins with the words "I'm sorry", which are not being used to express regret for one's actions. It basically says, "It is regretful that you feel bad because someone (who happened to be me) correctly pointed out to you that you are stupid, but the only way it can be fixed is if you smarten up."
Then there is this: "I feel bad that I called you stupid". Notice that while this is now an apology, it also does not say "I do not think you are stupid, and should not have called you that". It expresses no commitment not to call that person stupid in the future (and feel bad again). It basically says, "You actually are stupid, but pointing that out to you made me feel bad. I wish someone else had said it instead of me, and if you keep demonstrating that you're stupid, the same comment might slip out of my mouth again in the future."
It looks like Linda Holmes is criticizing the show 60 Minutes for making, in her view, a less than effective apology of some sort, perhaps along one of these patterns.