Promoting selflessness is a mistake, even if that is what you want.
An easy analogy to explain this is in the topic of Business Profits. In business, it is a big tactical mistake to focus on profits and keep trying to think of ways to make more profit. This kills more successful businesses than any other tactic.
This does not mean one shouldn't be aware about profits, they cannot be ignored: But it is important to realize that profits are a residue, what is left over from customer payments after satisfying the customers. The point of the business is to satisfy customers at less cost than the customers will happily pay.
The distinction seems trivial, but there are MANY easy ways to increase profits by making customers less happy: less choice, less service, less quality with cheaper parts, less guarantee, a worse return policy, and so on.
It is very difficult to make more profit and keep the customers equally happy. Most businesses have already minimized their staff and other expenses, so to make a bigger profit they turn to shortcuts, and lower quality (sometimes by overburdening their work force, which grows stressed and resentful and less attentive to the quality of their work).
The cost-cutting approach can work for many years, but eventually the drip, drip, drip of customers leaving (along with their best employees, because those are the ones that can easily find easier jobs) puts them in a financial bind, and the lost customers and good employees seldom come back: They have found something they like better.
It is more difficult but usually possible to make customers more happy, with a small increase in price and a smaller increase in costs, and thus take home more profit. Or if you have happy customers, to work on marketing to get more happy customers without changing quality.
So back to the selfless! You have a similar problem here; selflessness is, like profit, a residue. It emerges naturally from human nature when their own self feels satisfied: sheltered, healthy, safe, secure, socially connected and free to help. Or not. They do not have to feel rich. We have a lot of people in this position in the world: Retirees in countries where they receive a secure stipend they do not have to work for, along with health care they don't have to work for. Many of them donate time and energy to various causes.
I would warn that numerous psychology experiments show that rewarding somebody for any act (which includes selfless behavior) tends to make them think transactionally. In particular this includes behavior they did for free (like painting, or playing a game) because it was fun: If they start getting paid for it, even \$1, they tend to no longer do it for fun: They expect a dollar, and doing it for free is no longer fun because it isn't rewarded! This is true for both children and adults.
The answer is, if you want to promote selflessness, you should do that indirectly by promoting the kinds of conditions that eliminate the barriers to selflessness most people have. What are those? Trying to make sure they don't end up homeless, trying to make sure their kids are clothed, fed, educated and have healthcare, trying to make sure their own retirement is not going to be a stressful hellscape at the very time their physical and mental capacities are on the decline, and their health costs are increasing.
Now it sounds like I am advocating for socialism, but I am really not (at least not the socialism most people think of). The point isn't to make all of society equal, or equally rich. The point is not to put a ceiling on anybody; the point is to put a very substantial floor on how far one can decline financially: That floor is you will always have enough to not go hungry, enough shelter and warmth/cooling, water and sewage services, enough safety to be seldom worried about victimization by criminals, enough healthcare to not worry about suffering for lack of care, and enough education for your children to not worry if they will be able to make it in life.
If you promote that, selflessness will be maximized automatically.
Added to address OP commentary.
Social status will always matter; that is human nature.
Power does not always matter. I provide a specific circumstance, in the hope you can generalize to the broader culture:
At the university I had dozens of friends I got together with in various combinations, but none of us were ever "in charge" or deciding what we will do, where to eat, what volunteer projects to join, etc. New professors and college workers were often invited along, usually to lunch or a presentation or whatever. They could become our friends and be part of our informal group. But we would quickly disinvite and rebuff any that insist upon trying to be "the leader". We don't want leaders, even our department head was elected by us and could be un-elected and replaced. We want colleagues, companions, and people with a sense of humor, and the same goes for collaborators on our scientific projects: Nobody wants to spend two or three years of their life working alongside a jerk that is constantly trying to assert they are in charge and always knows best.
That is the part of the point of having a social connection first; to see if people can go along with somebody else's idea, even if it just Thai food for lunch instead of Pizza. They don't have to like it or pretend they did, but if they can't shrug off a minor mistake and all they do is complain about how Pizza would have been better --- In our culture, that personality trait goes beyond lunch and makes them a solo researcher in their career.
The same would go for trying to insist upon a place for lunch: If Richard asks me if I feel like Thai because he knows I like it, and we invite newbie Samuel along with us and he doesn't like Thai, he can decline without penalty. If he wants burgers, we will say we are set on Thai, if he tries to insist we should try his burger joint, we still refuse and again, that attitude of not taking "No" for an answer, if persistent and repeated, will result in academic isolation.
Now to apply that to your culture at large: In a post-scarcity society (a mini-version of that is approximated by academics and researchers working in a university setting), the notion of Project Leaders is acceptable and often an organizational necessity. Somebody has to pay attention to coordination and project-wide decisions, funding, schedules, progress and information dissemination.
But Power over others does not have to come with that; such power (for example to fire somebody, hire a person or firm to do something, spend large sums of money, or rewrite the budget) can be reserved to various committees by majority vote.
What makes people act selflessly, even if they are not inclined to do so, is a culture that rewards them with friends and collaborators for what they care about; and the punishment of isolation if they show a pattern of acting selfishly or trying to gain power: No friends, no colleagues, no collaborators. This is not a monetary reward or punishment, but even more effective than that.
The post-scarcity part is very important here: The strong safety net means nobody must work for anybody, so they don't work for people they do not like, or regard as selfish power-mad jerks. IRL that type of person gets away with their behavior because so many of their employees feel trapped in the job and forced to put up with a boss they dislike, or even hate. With a strong safety net, young adults won't tolerate such behavior: They in particular have little to lose (in terms of lifestyle) by quitting and falling into that safety net, so they just won't take jobs (or stay in them) if their boss is overly controlling or doesn't treat them fairly. That in turn means very few such people ever get to be bosses, because their turnover rate is too high, constantly incurring new training costs and learning curve, which is too expensive for their bosses to remain competitive.
That is cultural feedback: Bosses need approval and friendship of their subordinates, and that same personality trait has to eventually travel up to the top of the pyramid; because nobody on the bottom layer of the pyramid (our minimum wage slots) has to put up with any shit and can walk out at any time with zero consequences in terms of their lifestyle.
About 97% of people want to have friends, and the big, society-wide projects that bring high social status require teamwork. You can't make an action movie alone, you can't build the Large Hadron Collider alone. You can't even build much of a business alone, if the culture of working with friends that share equally means few people want to work for you!
Of course one can write a novel, or a song, or a mathematics paper alone, but for most life is more enjoyable with friends and co-workers, and that is also the route to getting bigger projects done that carry more social status. It is nice to be a part of something, and held in high regard by your friends. If much of that regard demands you be a good team player and selfless, then that is what you do.