Bureaucracies tend to take on a life of their own. After some years of existence during which the original intent of the organization may simply disappear with other remnants of history, the bureaucracy that was created for one purpose or another simply morphs into … whatever else will be acceptable to those paying the bills.
This has been documented innumerable times within agencies of the US government, and applies just as readily to nearly any government entity that depends on public funding. (It happens more often than we’d like to imagine in private industry, too, but with the management and bill-paying being generally better-connected than in government, those conditions generally don’t last so long.)
When it started out, of course, NATO was a purely military organization (well, to the extent that anything in any human organization can be free of politics, but at least political ambition and maneuvering played only a supporting role to the military one). Since the early 90s when we consider that “the Cold War ended” with the collapse and reorganization of the former Soviet Union and Eastern Bloc, the military mission has definitely become secondary to the political one.
The political ambition is … not so far off from what @ragingloli suggests. It’s a way for the USA to remain intimately connected with vital strategic decisions within Europe. Because even if the Cold War is over … Russia has for centuries had eyes on whatever warm-water access it can get for both economic and strategic purposes. In addition, Russia has for centuries viewed the plains to its west as one of its most vital buffer against military (overland) conquest, and for that reason becomes very nervous at the consideration of a western-aligned Poland, Belarus and Ukraine. If Hitler had been able to forge alliances with those nations instead of invading them, then World War II would have ended very differently than it did, as he surely would have taken over at least Western Russia to the Urals as well as the oil fields of the Caspian Sea.
In very simple terms, Russia wants to keep the countries on its western border allied with itself – or occupied by itself – to insure the safety of the Motherland itself. One of NATO’s delicate balancing acts is preventing outright provocation of the Russians by making the border countries hostile to Russia – as well as by preserving some kind of cohesion in European forces – and reliance on the US – in case a shooting war ever does start again.
One thing that NATO does, though no one wants to talk about it, is to provide reassurance to Germany that they don’t need to rearm themselves again – as they easily could – to ensure their own independence from Russia.