The alternatives to investment-based pensions do have some trade-offs. Both the tax-based and "public insurance" (current payers finance current pensioners) model are vulnerable to demographic shifts and economic depressions. In many countries with bad demography, the payout you get per income-month has been reduced and the age of retirement increased, because the system threatened to collapse. Since most of the world is now aging rapidly and living longer, these systems are under severe strain.
A significant advantage of pension funds is that they can invest abroad to spread the risk, but in a public system this is impossible.
small nitpick: Pension funds invest only tiny percentages into actual private companies, like the PE firms TFA is referring to.