It is only in the space of "mixed strategies" that there need be an equilibrium. And in many (most?) cases it is not at all clear how a mixed-strategy equilibrium might come about. Therefore Nash's result is more limited than it might at first sight appear.

The Economist's full article (behind the paywall) in fact repeats this claim that I consider misleading: "Nash showed that every 'game' with a finite number of players, each with a finite number of options to choose from, would have at least one such equilibrium."

An equilibrium that does not depend on the notion of mixed strategies is called a pure-strategy equilibrium. For an example of a game that has no pure-strategy equilibrium, see https://en.wikipedia.org/wiki/Matching_pennies.

Every one of such systems that are non-cooperative, and all players have perfect information (of rules, strategy sets, and associated payoffs).

Rarely does this happen in the real world.

Yeah which is a detail on prisoners dilemma. Again not the thing has the everyday importance.