If the aim of evolution is to allow an organism to better compete against rivals, why would stabilizing selection ever happen? If you're not selecting the most highly adapted competitors at either end of the spectrum then how would a species progress?
It occurs when a beneficial characteristic has been developed over time and it would be harmful to stray from it. In these cases it is not the individuals at the fringe (as you put it the most adapted) who are the best adapted.
I think it may help if I answer with an example that is widely promoted by AQA in their A2 Biology syllabus.
Stabilising Selection in human birth weight
It is harmful for an infant to be born with a very low birth weight. They are much more vulnerable to heat loss due to their high surface area to volume ratio and consequently their respiratory demands are very high. Pre-term babies (which account for 67% of low-birthweight infants(1)) are particularly susceptible to respiratory problems (lack of surfactant in the lungs), cardiac problems (Patent ductus arteriosus - the lungs are still bypassed when the umbilical cord has been cut) and dangerous intestinal problems (Necrotizing enterocolitis) amongst many other conditions can all be fatal (further information on mentioned conditions) and are reflected in high mortality rates at these low birth rates. It is therefore not beneficial to be on the extremes of birth weight.
Similarly, delivering a child of too high birth weight can cause complications with delivery if the head and shoulders are too wide to pass through the mother's hips. Therefore the other extreme of high birth rate is also not beneficial and will not be selected towards.
This leads to selective pressures in both directions, stabilising towards a mean birth weight as shown below:
This is an example of evolution not pushing a species forward but ensuring that individuals have the best chance of getting to reproductive age themselves.
(1) Martin, J.A., et al. (2007). Births: Final Data for 2005. National Vital Statistics Reports, 56(6).
Evolution has no aim, you seem to be repeating a caricature of what the theory of evolution really says. Evolution works by a mechanism where pressures on a population select for the traits that are best suited for that environment. If there are no pressures on a population, then the selection of specific traits will be slowed down, as all members of the population have an equal chance at reproducing.
Now to wax a bit philosophical on the issue.
"Survival of the fittest" is a conversational way to describe natural selection, but a more technical description speaks of differential rates of survival and reproduction. That is, rather than labeling species as more or less fit, one can describe how many offspring they are likely to leave under given circumstances. Drop a fast-breeding pair of small-beaked finches and a slower-breeding pair of large-beaked finches onto an island full of food seeds. Within a few generations the fast breeders may control more of the food resources. Yet if large beaks more easily crush seeds, the advantage may tip to the slow breeders. In a pioneering study of finches on the Gal pagos Islands, Peter R. Grant of Princeton University observed these kinds of population shifts in the wild [see his article "Natural Selection and Darwin's Finches"; Scientific American, October 1991].
The key is that adaptive fitness can be defined without reference to survival: large beaks are better adapted for crushing seeds, irrespective of whether that trait has survival value under the circumstances.
Living fossils (organisms that have not changed for millions of years) simply means that they evolved an adequate structure for a relatively static and unchanging environment, good enough to maintain a niche.
Or as a particularly brilliant writer that goes by the name of Calilasseia said:
The static species fallacy.
This is a particularly stupid canard, which the above discourse on inheritance, and variation brought about by meiosis, should flush down the toilet at a stroke. But, in order to reinforce how stupid this canard is, it is necessary to cover rigorously what a species is.
A species is a population entity, and as a corollary thereof, a dynamic entity. A species is defined in rigorous biological work, as a population of living organisms, whose members can produce viable offspring with each other, but whose members can not produce viable offspring with a separate, distinct population. Actually, this is only one extant definition, but it is the one that matters with respect to evolution, because once again, it points to the central role of inheritance.
Of course, part of the problem arises because of taxonomy. Because scientists need a reference point from which to launch further investigation, they have alighted, courtesy of our old friend Linnaeus, upon the process of cataloguing organisms and providing them with a unique, unambiguous identity. This, of course, has been most helpful in furthering our understanding of the biosphere, and indeed, Linnaeus himself, on the basis of comparative anatomy alone, alighted upon the idea that organisms were related to each other a hundred years before Darwin, which is why he constructed his taxonomic scheme in the manner he did. Yes, that's right, a creationist (though he was only a creationist because no other option existed in 1758) alighted upon the idea of biological interrelatedness, as a result of paying attention to reality. But the very same taxonomic practices that have been useful to science, have also led to a popular misconception. This is because taxonomists base their classification upon individually sampled organisms, one of which is chosen as a 'type specimen' that is henceforth declared to be the reference standard against which all others are compared. Other specimens are maintained in order to provide a record of likely variation in characteristics from that reference standard. The trouble is, of course, basing the entire classification system upon such reference standards promotes the illusion that those standards remain in place for all time. Scientists, of course, recognise that this is not the case, but it takes diligent intellectual effort to recognise that the taxonomic standards are merely particular snapshots of the state of the species at a given point in its history, which scientists then choose as their reference benchmark for current work. The species itself, however, courtesy of all that dissemination of variation across generations, does not stay still. It is NOT static.
I cannot reinforce this strongly enough. A taxonomic classification is merely a historical snapshot of the state of a species, used as a reference point for further work, and does NOT constitute "the species" itself. The species itself, is the sum total of all the living organisms comprising that interfertile population, and with each new generation, that population undergoes change, because in the new generation, each of the organisms comprising that population are genetically different from those in the previous generation.
So, if anyone wishes to erect the ridiculous idea that a species is a static entity, the simple retort is this. Look at your family album. Are you identical to either of your parents? No? There's your evidence for the dynamic nature of a species. Now replicate that evidence across millions of humans, and picture what happens with each new generation, remembering that across generations, inheritance is a dynamic process. There goes the static species fallacy.
I hope that clears it up for you.