The fact that changes how most people should think about this category: you already own it. Broad stock index funds weight their holdings by market value, which means each company's share of the fund is proportional to its size. The largest companies are, by definition, the largest share. So a total-market fund is predominantly a large-company fund with a tail of smaller ones, and an S&P 500 fund is a large-company fund outright. Someone holding either and then buying a large-cap fund "for stability" has not added an exposure; they have increased one they already had and reduced the small and mid-sized portion that was the only thing distinguishing a total-market fund from a large-company one.
Concentration is an output of the weighting, not a decision. When a handful of very large companies grow faster than the rest of the market, their combined share of a cap-weighted index rises mechanically, and the index becomes less diversified without any committee choosing that. The reverse happens when leadership broadens. An investor holding a broad index fund holds whatever concentration the market has produced, which is worth knowing rather than correcting. The way this shows up inside a specific index belongs with the S&P 500.
What size actually buys, and what it does not. Large companies usually have several products, several markets and several sources of financing, so a single lost contract rarely threatens the whole business, and their shares trade heavily enough that the gap between buying and selling prices is narrow. They are followed closely by analysts and covered constantly in the press, which makes information easy to find and makes it correspondingly less likely that any individual is the first to notice something.
What size does not buy is safety. Being large changes the way a company gets into trouble rather than whether it can, and a share of common stock in the largest company in an index is still a residual claim that ranks behind every creditor. Large companies also have the least room to grow proportionally, since a company already worth hundreds of billions has to add an enormous amount of value to double.
The named large-company index is not a definition of the category. The S&P 500 is often used as shorthand for large-cap US stocks, and it is close enough for most purposes without being the same thing: its membership is chosen by a committee against eligibility criteria rather than being the 500 largest companies, and it excludes small and mid-sized companies by construction. Different providers draw the large-company line in different places, so two large-cap funds can hold different lists.