Well, there have been other factors since WWII than the US trying to be world police.
It does happen sometimes in history that after a particularly destructive war, or series of wars, people settle down into relative peace for a time. The Thirty Years' War was followed by the Peace of Westphalia in 1649 - and certainly there were wars after that, but much less destructive ones for a century. Likewise the Napoleonic Wars ending in 1815 were followed by no wars between the Great Powers until 1850-53 with the Crimean War, but that was limited in geography, thus the name. After that there were no direct conflicts between Great Powers until the Great War - a period of 99 years of peace across Western Europe.
One of the things that comes out of these conflicts is some sort of mechanism where the Great Powers agree to maintain the peace, implicitly blaming minor powers for their previous conflicts. There was an idea of a Congress of Great Powers, which would meet from time to time to settle disputes, such as the carve-up of Africa. The League of Nations was a more egalitarian idea, with all countries having a single vote, and unanimous votes required to do anything much - but the United Nations was a return to the old idea, with the Permanent Members of the UN Security Council being in effect a Congress of Great Powers.
But this one sat permanently, they didn't wait for a crisis and then call one, they were always there. Thus a Soviet Ambassador to the UN was sitting in New York in 1962 during the Cuban Missile Crisis. When the Council met, the Soviet representative knew that if his negotiations failed, he'd be radioactive ash, too. This gave him a strong incentive to be polite and reasonable.
Having a permanent Congress of Great Powers means that instead of troops massing, you get Khrushchev banging his shoe on the table. Undignified, but less bloody.
And of course, all those Great Powers having nuclear weapons also makes a difference. It necessarily limits wars, so that the Great Powers are reluctant to enter into direct conflicts with each-other, and when they do get into conflicts, they try not to make them so destructive that people think nuking wouldn't be any worse.
There's been a growing humanitarianism, too. After the horrors of Dresden, Auschwitz, Katyn Forest and Hiroshima, the Great Powers, at least, decided to restrain themselves a bit. When the West invaded Iraq we did not start by firebombing Baghdad. When the Soviets invaded Czechoslovakia they did not simply shoot 50,000 people. Calley's massacre at My Lai was an horrendous crime for which he should have been hanged, but it was not an everyday occurrence, the US Army did not have Einsatzgruppen units as part of its order of battle.
US leadership most certainly deserves a lot of credit in reducing the severity of wars since 1945. But the other Great Powers deserve a lot of credit, too, and diplomacy in the United Nations is boring and bureaucratic and stupid - but has prevented a lot of misery. The UN is like vaccines, its very success makes us think we don't need it.