As much as I might like to disagree broadly with what you have written, I can't, because there is clearly a lot of truth to it from an evolutionary perspective. It's quite true that young people (teens, and twenties, especially, but also later as you point to) do try to show off in various ways to impress the opposite sex as part of human mating rituals. But, let me try to at least surround that truth would some additional options and nuances as a ramble.
First, as an example of a way to deal with this. In James P. Hogan's sci-fi novel "Voyage From Yesteryear" about a post-scarcity society, he addresses this by the notion that people compete to demonstrate excellence in their chosen skills. Showing excellence in helping the community become a form of "Wealth". Material goods are given away freely, including to those who make no contributions to society, in part because, if someone is "poor" (not contributing, so socially disrespected), why heap additional problems on them by not letting them have material goods? So, while you have outlined a truth, how society chooses to deal with that truth, how these urges are directed, is an aspect of culture and circumstance.
From another direction, life on this plane of existence seems to consist of both cooperation and competition, arrayed across a mix of both meshworks and hierarchies. As E. O. Wilson points out, organisms often cooperate within some defined social boundary (like an ant colony) and then compete outside of the boundary (like ant wars). Humans historically have cooperated within tribes, even as they fought other tribes to define essentially property line boundaries between tribes. Many people enjoy team sports where you cooperate in your team but compete against other teams. Even Genghis Khan's command organization must have had some sense of internal cooperation even as it may have attacked other communities. So, the healthy human brain is able to navigate this social landscape (at least withing historic boundaries and the "Dunbar's" number of 100 - 230 tribe members). So, again the issue becomes, how does society direct these impulses within the limits of human potential?
Freud had some keen insights, but he also overgeneralized and was a bit nutty. (People might say that about me, too? :-)
"Freud may have been bad. But can he really have been bad in so many contradictory ways? A sampling of recent books suggests that after a century of Freud flogging, the critics still haven't finished with him."
G. William Domhoff goes into detail about differences between the left and right:
One aspect not there is perhaps that the left tends to emphasize the cooperative aspect of society -- that we are all in this together, and if we all cooperate, we will all be better off, and that included caring for all children. While it may be rarely stated this extremely, the right tends to emphasize that people should succeed on their own merits, and part of success is being able to afford to raise children -- where if people can't afford children personally, they should not have them, and if they do have children, it is only right if the children suffer and die, because failure should not be propagated in order to maintain the health of the population.
There actually is quite a bit of sense to that sort of "Social Darwinism" from an individualist perspective -- except that it ignores both how much of success is collective, how sexual recombination crosses social rules about inherited wealth, and that the marketplace can be pretty fickle about choosing what "success" means (as opposed to when success as a hunter or gatherer was more obviously a sign of health).
Yet, undeniably, the process of evolution through selection every generation is, to a large extent, about constantly repurifying the genes of the population for what works (usually what worked in the past) and what doesn't work (usually mutations, with very rare exceptions). Note, by this purification, I don't mean anything like "Aryanism" or such ideology about "blood"; I am talking in terms of population genetics as someone who has been in a PhD program in Ecology and Evolution for a time, where "good" genes might, say, be ones diverse enough to resist parasites, and where "racial" boundaries across humans are essentially vague and arbitrary.
However, historically, a tribe that had overly lazy members ("free riders") who had too many children for the circumstances would not do well, so there are evolutionary problems with them left too. Our basic emotional brain wiring seems to reflect all that, even as different people may be tuned more towards one side or another. Even as most people can be generous and cooperative, most people also have an innate sense of fairness and also a sense of willingness to sacrifice or take risks to punish perceives slackers. Those are tendencies that have served tribes well in the past.
However, that does not mean those tendencies may be adaptive in a post-scarcity high-tech society. In such, it may cost more to ration things then to give them freely. Just one creative idea from one of a billion "free riders" will enhance the commons. Even one disgruntled person disenfranchised with nothing to lose might make a plague might kill everyone on a small planet (see the Elysium movie mentioned below for a variant on that). So, the whole social dynamics of the system may shift...
Apparently, historically, nasty uncooperative people have also been killed off every generation fairly regularly (with speculations of like 10% of the population each generation I read somewhere?). Things seem to have improved recently related to violence:
One theory is that in recent centuries under capitalistic-ish systems, less impulsive wealthy people could afford more children, and that lead to a spread of more self-control throughout the population over a span of many generations (related to the "right" argument above). Plausible, but not sure it is that significant?
As conservatives put it:
"This is no surprise, as [Propertarian] libertarianism is basically the Marxism of the Right. If Marxism is the delusion that one can run society purely on altruism and collectivism, then libertarianism is the mirror-image delusion that one can run it purely on selfishness and individualism. Society in fact requires both individualism and collectivism, both selfishness and altruism, to function."
Humans may be half-way between war-like Chimps and peace-loving Bonobos as to behavior:
Anyway, so I can't disagree with your fundamental point about human inclinations -- but I can say that culture affects how those inclinations are expressed and amplified.
In one analysis I read about why people like Clinton or Blair have affairs that may risk their political power, the point is made that historically the whole point of political power was, evolutionary, to have affairs (whatever legal or social principles may operate today). Which is essentially your point.
A couple other tangential points.
If someone like Ghengis Khan was so successful as to have 100% of males being his descendants (not the case, but consider), then suddenly there is nothing special about having his genes. So, suddenly cooperation would make more sense across all those copies of the same genes. Those common genes might even provide a new platform for evolution of diversity (see Manuel De Landa's "Meshworks, Hierarchies, and Interfaces"). So, there may be self-limiting aspects of a "selfish gene" idea. Sort of like, after rogue nanotech makes the wold into grey goo, or after the 1% wipe out the rest of humanity, then everything goes back to square one and evolution again.
Lorenz may have been brilliant as understanding animal behavior, but people do have many unusual characteristics, given the idea of culture, that go beyond genes (and Lornez's notions of genetics may have been biased by Nazi politics).
"I was frightened -- as I still am -- by the thought that analogous genetical processes of deterioration may be at work with civilized humanity. Moved by this fear, I did a very ill-advised thing soon after the Germans had invaded Austria: I wrote about the dangers of domestication and, in order to be understood, I couched my writing in the worst of nazi terminology. I do not want to extenuate this action. I did, indeed, believe that some good might come of the new rulers."
Also, there are many evolutionary strategies to reproductive fitness. Examples:
Diversity tends to be useful in survival. And female selectiveness in mate choice is one of those variables.
Anyway, so there is a lot of complexity to this once you start poking at it.
On "pushing people down", contrast being told "you're smart" with being told "the brain is like a muscle":
"It didn't take long. The teachers who hadn't known which students had been assigned to which workshop could pick out the students who had been taught that intelligence can be developed. They improved their study habits and grades. In a single semester, Blackwell reversed the students' longtime trend of decreasing math grades. ... While we might imagine that overpraised kids grow up to be unmotivated softies, the researchers are reporting the opposite consequence. Dweck and others have found that frequently-praised children get more competitive and more interested in tearing others down. Image-maintenance becomes their primary concern. A raft of very alarming studies -- again by Dweck -- illustrates this."
By the way, on 1%/99% conflicts, a new movie coming out called "Elysium" (although it glorifies the way of the warrior not the peacemaker):