To believe otherwise is to live an unexamined life, and the unexamined life is not worth living.
Some people feel like being altruistic, but feelings are the products of ideas. Ideas can be examined and thrown out when irrational (as in this case).
For another thing, it's not true that people are doomed to ignorance, and it seems like your argument probably rests on that assumption.
I don't see for a second why you'd say people are doomed to ignorance. We use sense perception to perceive stuff, and we develop a coherent picture of the world (so we learn to reconcile optical illusions, for instance). And for all intents and purposes, that is reality, and it's objective.
Maybe there's some other reality underneath that, but then, maybe there are pink unicorns dancing on Alpha Centauri; it's totally arbitrary, and thus irrelevant.
At the end of the day we all are robot executing a program.Yet saying that serves little purpose. It's as stupid as saying "we're all made of atoms", it doesn't make someone who is pain feel better. There are reasonable solutions today to most problems of mankind. Making a difference is important because we are all on the same "boat". No man is an island, even nihilists like you aren't.
Nihilism often goes unidentified. Anyone who has ever changed beliefs has acted momentarily as a nihilist. A person who is open-minded is a partial nihilist. A complete nihilist being someone who has developed a core belief that none of their other beliefs are certainly true. They simply recognize each as subjectively and temporarily optimal. Notably this evaluation changes, for every existing belief, every time you change any single belief. At it's extreme nihilism can be anxiety inducing. There's always that sneaking suspicion that your core belief is no longer subjectively or temporarily optimal. That and no way to test your suspicion but to maintain your core belief. You can get sort've stuck in an anti-dogmatic loop frantically evaluating philosophies. This doesn't have to be your experience of nihilism. You can simply adapt beliefs to manage this problem.
If you "choose to believe" something, it's just a fantasy. If you "choose to believe" something, you don't really believe it. To really believe something, you have to have enough evidence to be certain about it.
Fortunately, pleasure is inherently valuable (this is given by biology), and moreover, our brains supply us with pleasure (in the form of brain chemistry). So now we have a basic value that we can erect other values around. Of course only if they are valid. I can value preserving my life (for instance), art, my loved ones, and so on, but I can't value something arbitrarily.
(The exact psychological mechanism of genuinely enjoying and thus valuing art and loved ones, and such things, is not straighforward, but it's there. Seeing how preserving one's own life is a value is more straightforward. I am a total egoist, by the way, because you can't out of that with the framework I am describing.)
edit: In other words: you should not just choose to "believe" one thing or another. You should look at reality and decide what the evidence favors.
Otherwise, you don't really know, and you know that you don't really know, and you are flying blind in life, and thereby engendering in yourself a sense of helplesness and anxiety (no coincidence that one of Sartre's works was called Nausea...).
Second, if we want to make the world a better place, the best way to do it is to spread rationality and to counter forms of irrationality---in this case, altruism.
I'm not against human happiness. I'm for it. But I'm for doing it in a way that works, not a way that doesn't.
That does not sound like a good strategy.
It may lead you to being a totally self-sacrificial shmoo living in a fantasy world, like the woman profiled in the article. Or it may lead you to being a jihadist. Or it may lead you to advocating political ideas that are destructive (like, say, the Nazis).
Man has the faculty of reason, let's use it.
There is such a thing as empathy for other people or animals that is genuine, but you shouldn't sacrifice more important values for it. Similarly I like playing board games sometimes, but I don't choose to play them all day and impoverish myself.