It's a common criticism of Spencer's social Darwinism that it misunderstands the entire theory of evolution and the meaning of "fitness" such that it is commonly exploited to fulfill specific hateful political agendas. However, what exactly is wrong with the notion that the most competent individuals in society thrive? Consider an academic setting with a particularly rigorous professor running a particularly rigorous course. Obviously, only the most studious and tenacious students will be able to pass while the lazier ones will either fail or drop the class, and rightly so; only those who earn their keep should be rewarded. How is this idea immoral? Perhaps I am misunderstanding social Darwinism.
Biological evolution has nothing to do with "survival of the fittest". The hallmark of Darwin, Wallace, et. al. was the removal of teleology from biology. "Social Darwinism" - whatever that may mean to you or to Spencer - is simply a way of looking at society and explaining social events teleologically (in terms of goals, purpose etc.). Note the difference between these two explanations:
1) plants photosynthesize in order to survive.
1) this plant is photosynthesizing.
2) plant species which photosynthesize have an increased likelihood of survival as a species.
Note that survival is still a factor, yet in the former case survival is a presumed goal and in the latter simply a result.
As for "social Darwinism" you need to examine "fittest" - fittest according to whom? Thrassymachus? Simmons? Spencer? LaLane? You?? Similarly with "most competent" and "thrive" - it is not a question of morals, it is simply a matter of distinguishing what is true from what is "true to you". There is nothing "wrong" with "Social Darwinism" but there is also nothing true about it - only opinion to be either agreed or disagreed with like so much gossip.
In Darwin there is no abstract "fitness": evolutionary traits are fit, or unfit, for a given environment. Being white and very furry makes one fit for survival in the Arctic, not so in equatorial jungles. Besides that, it is clear that evolutionary traits must be inherited, or their selection would bear no consequences. Whatever it "takes" to be "successful" within a human society, there is no reason to believe they are inherited traits. And inequality within human societies presupposes the successful reproduction of the "unfit", too: if only the mighty capitalists survived and reproduced, who would toil for them in their factories?
Plus, Darwinism is a theory about, well, the origin of the species. It pressuposes, as such, the existence of many different species, which compete for survival in natural environments. But human societies are by definition composed by only one species; how would one discuss the "origin of species" in a society that has no different species to begin? And for such reason, social Darwinism cannot ever properly define what it is talking about. Is it about the survival and procreation of human individuals? In such case one has to admit that the poor are much more "successful" than the rich, because they make up, what, about 90% of mankind. Or is it about a completely different thing, the economic success of individuals and companies in capitalist markets? If this is the case, then it completely fails to acknowledge how things happen. Companies do not reproduce in a biological sence; whatever traits make them successful in capitalist competition cannot therefore be inherited. Human individuals do reproduce - but "what it takes" to be a successful capitalist doesn't seem to be inheritable at all. Indeed, even the common sence folk lore knows that: "rich parents, noble children, poor grandchildren".
So, the relevant levels in biology do not find adequate equivalents in social science. In biology, there are genes, subcellular structures, cells, individuals, species, and biosystems. "Biologist" theories of society cannot accomodate those different levels to a different subject. If society is the equivalent of biosystems, what is the equivalent of individuals? Or of species? That, precisely, is where social darwinism fails. Economical competition among humans in a human society is extremly different from biological competition among animals of diverse species in a biosystem. Hence the flaw.
My view of "What exactly is wrong with the notion that the most competent individuals in society thrive?", is that if a few individuals hold more than 50% of power in a society they would have the ability to suppress or remove the remaining individuals. The moral side of this is that if an individual rises to power more quickly than everyone else they may have the most power at a given time but it may not be the best possible scenario for all individual's involved.
A society where everyone earns his own keep is not so different from the one in which each man hunts his own food. There will be no leisure, no civilization, no progress, no great art, no great science, no great philosophy because everyone is purposeful, and everything is done with a profit in sight. Unfortunately, the ultimate source of creativity is blind impulse.
Civilization was created by the leisured class: in the west they were slave owners; in the east, they were plain old parasites (or eating guests). The slave owners mused for pure pleasure; the parasites pondered for their masters, and thus was more or less purposeful. As it turned out, those who mused for pure pleasure created the greatest civilization.
A society where everyone is on the fringe of survival is a very nasty one. People will lie and cheat, will hit high and low and stab both in the front and in the back because it is their lives that is at stake. To have a sense of what it feels like to live in such a society, try to think of some insecure corporate employees who treat workplace as a survivor reality show: they strike low, and they strike first, always.
Nevertheless, social Darwinism has never ceased to be in operation, as a consequence of which Greek intellectual giants were thoroughly exterminated from the gene pool; this explains why Byzantine, with all the books and scholars, had remained stagnant for a thousand years.
Since I do not believe all men are equal, I do wish people with certain traits could have been more prolific, not on account of their ability to earn a living, but in virtue of their contribution to civilization. Da Vinci is one, Galileo is another, and the likes of their parents and great parents all have my best wishes. As of today, social Darwinism has been very cruel to this class.
There's a few issues which show up when talking about Social Darwinism.
The first major issue is the definition of "fitness." When it comes to species, we are comfortable defining fitness in terms of the ability to reproduce. However, when applying it to people, we tend to use the more general meaning of the word. This leads to the typical attempts to use Social Darwinism to justify superiority. I'd point to the film Idiocracy for an excellent counterexample to this typical usage. In that movie's plot, intelligence leads one to have fewer children, so we see evolution preferring the less intelligent who could reproduce not only their genes, but their ideas.
The second major issue is that societies are rarely simple, making it harder to identify the actual selection taking place. It's hard to tell the difference between someone who is "more fit" vs someone who simply got a life that was more conducive to being fit. This is the argument with "white privilege," where it is argued that white people at the top of society aren't more fit, they're simply delt a better hand in life. This makes it harder to use social darwinism to create any meaningful predictions.
I am certain it is reasonable to apply the theory of natural selection in societies, but doing so is simply difficult. Society makes it harder to determine when fitness is the major factor versus luck of the draw. This makes it very difficult to draw any meaningful predictions. When we do try to make predictions, we tend to use weaker definitions of "fitness." These weaker definitions cause all sorts of improper conclusions.