Sensitivity largely means treating others as fully human even when they're upset by something that doesn't or wouldn't upset you.

Key rule: you don't need to empathize with everyone all the time. Empathy is yours, and theirs. It isn't someone's to command.

There's a little twist, though. "I don't have any sympathy for..." is an apparently mean stance common among people who are not usually mean. It seems to translate to an irrationality: "Because I am not going to invest in feeling for you, I don't give a shit about you." (It really isn't either/or like that.) Personally, I recommend removing the phrase from your social field guide and finding other ways to talk and think in those situations.

You'll find it clarifying. You aren't required to feel sympathetic or empathetic, but you might want to ask yourself what "I don't have any sympathy for..." actually means, or establishes, in your world. Listen to your own tone. When I hear those words, they always seem to serve as a conduit of willful ignorance, an insistence on a mental block. It often seems to be an expression of contempt and a dismissal of an entire person; not "I'm not feeling empathy, here" but "this person deserves for no one to understand them." The phrase strikes me as a somewhat lazy rationalization for just plain misunderstanding.

Let's consider a parallel... Listen, swear words don't make you a bad person. Not at all. But if you find yourself swearing every other word and calling all manner of items "shit" ("Where did you move my shit?") or "stuff" or "things," you might want to put the kibosh on that temporarily and use the rest of your vocabulary. If you couldn't say "shit," "stuff," or "things," what would you say?

Likewise, if you couldn't proclaim that you have no sympathy (not so subtly implying no one should), what would you say? Ask yourself whether the question of sympathy or no sympathy might be a distraction from a more important question. Does a person need your sympathy or empathy before you will see them as a human being who makes sense to themselves? Are you relieved of the requirements of decency when you decide you don't feel sympathetic? Can you learn something useful from this scenario regardless of how you feel about a particular person or their role in it?

Sensitivity earns its keep when we don't understand, don't feel sympathetic, don't believe we'd say or do the same.