Ayn Rand properly recognized that people do not necessarily act in their own self-interest. There is a difference between having a psychological motive to act in a certain way and a self-interested reason to act in a certain way.
Having a psychological motive can simply mean having a subjective whim, and an action taken on this basis can be very damaging to oneself (and so, not self-interested.) There is also self-sacrifice out of a desire to “be good,” i.e. to follow the duties of an irrational morality.
(Imagine a young woman who has the passion and ability to become a great artist. She wants to become an artist, has ideas for great paintings, and being an artist would be her means to a happy, flourishing life. But she accepts the morality of altruism, and she is convinced that, rather than going to a top art school, it is her duty as a “privileged American” to devote her life to saving Third-World children. This is not self-interested, just because she wants to save Third-World children out of a sense of moral duty. Nor is it self-interested just because she may get some pleasure out of “being moral”–at first.)
Given one’s basic nature, situation, experiences, abilities and psychological makeup, one’s own self-interest is objective, not a matter of one’s momentary whims. Any pleasures one pursues must be consistent with one’s overall, long-term well-being–both physical and mental–to be moral.
Those who refuse to think and continually subvert their own minds are not selfish. They need their minds–their ability to reason–to achieve any positive thing they value. What can they achieve without reason? In principle, only their own destruction, and, to the extent they use force, that of others.
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Atlas Shrugged, Altruism and Egoism
Why a Proper Ethics is Not a Set of Social Rules, But a Complete Way of Life
Values Are Relational But Not Subjective
QuickPoint 1: Thinking is Individual
Human Emotions are the Products of Beliefs and Subconscious Value Judgments
Objectivism In Depth 
Sorry, but this seems a bit superficial, to me at least. Seems like the author didn’t dig too deep before denying this thesis.
“This is not self-interested, just because she wants to save Third-World children out of a sense of moral duty. Nor is it self-interested just because she may get some pleasure out of “being moral”–at first.”
What is “being moral”? What is “a sense of moral duty”?
It’s the values we’ve been taught and developed over the years, the sense of idealism, the image of an ideal self situated in the Superego. You can use Freud’s model of division of the psyche to explain almost everything.
So, if I had been taught to get good grades at school, I shall feel guilty if I don’t, because the Superego will punish the Id-the drive to do the things we really want to do deep inside, notwithstanding the influences on thought.
The same goes for moral values, if I had been taught to help people, I shall also feel guilty if I don’t.
The whole idea is based on the concept of gaining pleasure or avoiding pain.
To avoid pain, we act in ways we’ve been taught.
And that’s the source of selective empathy-we empathize with things we could link our own selves with, that is, with things we had experienced personally or situations we know to be painful vicariously.
When people empathize with things they have neither experienced nor have been made aware of, simply things that they do not understand at all but have been required to empathize with in some way (which is very rare), they do it because of a learnt value established in the Superego-the value of giving space to the unknown. In my personal experience, a friend did that for me-“I don’t understand this particular experience of yours, I’ve never been through that nor have I heard of anything like it before, but I see that you’re troubled and I empathize with you”.
And why did she do this, or to be more specific, why had she established such a value in the first place?
Because every person finds themself in a situation where they feel detached and misunderstood by the world, so, once again, selective empathy-we empathize with other people who feel bad even if we don’t understand their burden because we had previously been in the same place.
And this is quite an interesting “circle” actually-we empathize with something we don’t understand because the very “misunderstanding” is our own weak spot as well!
There are many, many other examples that substantiate the thesis.
Take “overcaring” parents for example (the most common type, and that’s no coincidence). If the parent calls their child non stop while they’re on a hiking expedition for example, even if their child is very experienced, mature, and there are all the objective reasons not to be so worried, it is because the parent fears for themselves, they want to avoid the pain they would feel if something were to happen to their child.
So, what people call “great care” is actually selfishness.
And in my personal experience, my mother has always been the loving, the emotional, the caring one but simultaneously a very conscious, intelligent person, and my father has always been a careless brat who has thought of nothing but himself his whole life and yet, when I go mountaineering, he is the one making all the drama rather than my mother. Because my mother knows how to distinguish between subjective and objective and knows how to cast aside the subjective and decide rationally, while my father ignores all the facts and makes the fuss because he wants to avoid his own pain, he wants all the pieces in his life to be in the same spot, gathered together, which is selfish.
Now, the fact that my mother doesn’t bother me and yet she does care a lot, is once again psychological egoism-she doesn’t want to irritate me with stuff I deem to be unnecessary (and have every reason to deem it as such) because if she does she will disturb my peace and well being, and she will feel guilty about it afterwards. It’s all connected.
Then: why do animals only care for themselves, and if they care for somebody else, it is because they benefit from that other animal or person? Or, observe babies as they grow up. They have no “sense of morality”, the only thing they feel is the natural urge to gain pleasure or avoid pain, and they act in ways that would assure these aspects, and they simply learn those ways from their parents, they learn that if they keep touching their phones they will either be in pain or devoid of pleasure (“if you keep touching my phone, I won’t buy you your favorite juice”). That’s simply our biology, our nature (infantile narcissism). Nature has made us “selfish” in order to survive. If there was actual, real goodness, without any obvious benefit or the benefit of simply feeling good because we did what we thought was right, we would destroy ourselves. We evolve and build defensive mechanisms of all kinds so that we could repel the outer influences of any kinds (both psychical and psychological).
The problem with this thesis is simply the name. When people hear “egoism, selfishness”, they think of all the bad things in the world and they instantly build a wall against further explanations, it’s another defensive mechanism. I’ve tried discussing this with people and they wouldn’t even hear me out, and this is an immensely vast subject.
Hence, I think that many terms need re-defining. The terms we use to describe certain things in life are not what those things actually are.
Love? What is love? Love is simply another link between people on the basis of self-interest. We love people who give us what we want to receive, it’s that simple. If those people are different from us and we still love them, it could be because of many reasons-the fear of being alone and not getting all the good things we get from that person, a value established in the Superego that tells us that we must accept other people as they are and so forth.
If there was such a thing as “goodness without reward”, then how come we cannot just love anyone in the world? How come we are so picky about our partners?
This requires deep, deep introspection to cognize, and a great deal of courage, because it is indeed a delicate subject, nobody wants to believe they’re inherently “selfish”, right?
But my point is that, we don’t necessarily need to call it selfishness, self-interest or egoism. Because it is a “healthy” type of selfishness, the one that makes us survive in the world, and we must embrace that. Only then will we become completely free, without judgement towards people and so on, because we all have our own, different needs and fight different battles, and when different battles and interests collide, a conflict arises. The deeper I get into this thesis, the freer I become, because I no longer judge people, rather, I understand them as I understand myself, even when they do me “bad” (another term we need to redefine. What is “bad”? Something universal, or another person’s selfishness that collides with our own selfishness? We are often very hypocritical here).
Now, finally, there is the question “how does the world work at all if we are inherently selfish?”
On the basis of alignment of the self-interests. Because we work in such ways that our own interests (again, desires for well-being or avoidance of pain) are aligned with those of the world. If you empathize with me when I’m in trouble, I shall empathize with you too, and it will work for us both, though we’re doing it for ourselves. You selectively empathize with me because my condition resonates with your own experience or understandings and makes you remember how bad you felt when you lived through the same/similar experience, or because of the other reason explained above. And I empathize with you for the same reasons, or to simply repay the favor (another value from the Superego, repaying favors-right?).
Because our “selfishness” can work in such a way that, while aiming for one’s own pleasure or avoiding one’s own pain, we end up behaving in such ways or doing such things that are useful for other people too, as a by-product of the initial motivations.