“is there any war the US engaged since the formation of the NATO alliance...”
The purpose of NATO was not to help the US to wage war, it was to provide deterrence against other countries attacking any of the NATO countries.
Actually it was organized after WW2 for the US to provide deterrence against the old global monolithic threat, the USSR. The USSR exited the scene in 1991 and The Russian kleptocracy succeeded it. Continental Europe has a population more than 3 times larger than their only remote regional threat, Russia, and a GDP 10 times greater That is when we should have taken the France route and exited Article 5 not committing ourselves to rote military action on their behalf and, as France did, merely subscribed to the joint diplomacy committee. Other than the other 4 Anglo nations, who have always pulled their weight in NATO actions and should be the only nations we have a mutual defense compact with, never has the contribution by continental nations been material to wars in the Middle East or even on their continent in the Balkans which they hounded us to get involved in. We can supply them intelligence and sell them our advanced weaponry but that should be the extent of our contribution to a continent reluctant to defend itself. That should light a fire under them to raise a joint continental standing army
The purpose of NATO was to keep the Russians out, the Americans in, and the Germans down.