PR folks have mastered the art of the “non-apology,” and we are all the worse for it.
First of all, “I am sorry” is simply not an apology. It is completely ambiguous, with two meanings that in context are often contradictory.
On the one hand it can mean “I apologize,” which is relatively clear, unless qualified by words like “if anyone is offended.” But “I am sorry” can also mean, “”I sympathize with your feelings, which may or may not not necessarily agree with the underlying source of those feelings. So, the speaker gets off the hook without admitting the problem.
Spanish has the wonderful phrase “lo siento.” “I feel it,” which directly addresses feelings. I would suggest that we say either “I apologize,” or “I sympathize,” depending on context. Not that either are perfect, but at least in the right direction.
Secondly, even if the words formally indicate a true apology, I do not believe them, at least from public figures, unless they are accompanied by an explanation of an understanding of where the mistake came from and of the damage it has caused, as well as what the person is doing to prevent it from happening again.
For example, when a public figure uses a hurtful word, the explanation might include an analysis of the speakers feelings, and of the complexity of the way they use the word, as well as how the deeper understanding they have now reached will lead to more careful thought with all language.