The key mechanism of Adam Smith's capitalism is a self-interest which manifests itself systemically as The Invisible Hand. It's a concept that offers a wealth of self-satisfaction: by pursuing our most selfish interests, the whole of society benefits.While this is certainly the dominant narrative of what the Invisible Hand aphorism means, it bears little resemblance to what Adam Smith actually said in the full context of when he said it.
By the twentieth century Adam Smith was largely forgotten and rarely discussed. He and his legacy had been eclipsed by economists like Ricardo and Pigou.
In their pursuit of a new, revitalized form of classical liberalism, the founders of neoliberalism resurrected Adam Smith and claimed his legacy as one of their guiding lights. Walter Lippmann was first to do this in 1937 in The Good Society. Henry Simons followed suit in the 1940s with a series of essays that were collected and published in 1947 as his Economic Policy for a Free Society. Simons appears to have been the first to directly invoke Smith's Invisible Hand aphorism, but he did so very loosely and, arguably, faithfully.
While it is not clear when the current narrative of the Invisible Hand was created, its greatest and most ardent peddler was Milton Friedman. Friedman was much more than an academic, he was a very public and articulate personality that once had his own television show and wrote several books.
Margaret Thatcher bought what Friedman was selling, and proudly proclaimed "there is no society."
Adam Smith would have been shocked and appalled by Thatcher's statement. After all, before The Wealth of Nations, he had written his Theory of Moral Sentiments, which was all about the ethical standards and codes of conduct that bound a society together, and it was in the Theory of Moral Sentiments that he first made reference to the Invisible Hand.
Indeed, Smith's Invisible Hand was the social code that bound individuals together into something greater than the sum of its parts-- society-- and guided their conduct towards the common good. In Smith's time, the individual truly was the engine that fueled the economy. While there were limited liability corporations at the time, they were creatures of the sovereign that lived and died at its whim. The vast majority of business was conducted by human beings who were fully liable under the law and personally accountable under the social code for everything they did in business as well as in their personal lives. Given the fact that businesses run by individuals were already fully accountable to and guided by their position as a member of society, there was no need to impose further restraints on trade.
In his 1930 The Political Element in the Development of Economic Theory, Gunnar Myrdal referred to Smith's notion of a society unified and guided by a harmony of interests as a "communistic fiction," noting that it was "implicit in most writings on economics" and a staple of economic liberalism. In this sense, as noted by Hannah Arendt in The Human Condition, classical liberals like Smith, Ricardo and J.S. Mill created communism by essentially asserting that promoting the common good was the goal of laissez-faire.
When it proved that laissez-faire did not, in many cases, promote the common good, it was natural for somebody like Karl Marx to reject the individualism of laissez-faire as the fiction and embrace the communistic fiction as reality.
Where Marx excised the "individualistic fiction" of classical liberalism, the founders of neoliberalism excised the communistic fiction. To their minds, the true failure of classical liberalism was including the communistic fiction in their economics and then mistaking that fiction as the goal of economic policy. To the founders of neoliberalism, laissez-faire was the goal, whatever its consequences to society.
Smith's Invisible Hand aphorism proved very malleable because it is stated as an identity: individuals pursuing their own selfish interests automatically promote the common good. It was a simle step for somebody like Milton Friedman, whose economic "theory" is itself merely an identity (the quantity theory of money), to provide the addendum "so don't worry about the common good because it happens automatically."
The fact remains, though, that Adam Smith believed that the common good of society, not self-interest, was the goal, and that the Invisible Hand exercised by common "moral sentiments" of individuals within the society would guide them toward that goal. In Adam Smith's Invisible Hand, as in nature, the individual and society are inseparable.