At no point in history was Rome, the Mongols, or Britain a global hegemon or even extant in a unipolar world.

The entire existence of the Roman republic and empire was bookended by China, for example, and Rome was almost continuously at war with almost all of its neighbors. Some of whom repeatedly beat them and eventually definitely outlasted Rome.

The Mongols defeated the vast majority of entities they went to war with, but not all. This is the closest you'll come to your thesis, as it is probably true that no army or civilization existed from the Pacific to Atlantic oceans capable of defeating the Mongol empire at peak. It is alt history to debate what might have happened had they pushed to Paris etc. They didn't bother. They easily could have, in my opinion, but had other priorities in Asia.

However, the Mongols were definitively repulsed in the Middle East and generally speaking were not going to conquer Egypt or Africa. They didn't even try to hold Jerusalem (though interesting quirk of history they took it, once). The Mongols were also arguably repulsed by India, though that's complicated by their own factionalism (they arguably defeated themselves in a civil war at the border of India).

It is probably true that the Mongols occupied any territory they chose to, at the peak of their power. However, they often chose not to, either because they acknowledged the impossibility of doing so, or the complex multipolar reality within and surrounding their empire required their presence elsewhere to maintain power. So that is also not a hegemon, but is probably the most terrifying and powerful empire in history. Certainly the only one to execute all civilians in conquered cities, with knives, in one day. Repeatedly. For sure the Mongols out-murdered anybody ever until Stalin and Mao. But this is not so much unipolar as systemic genocide. The Mongols literally used genocide not as an end but a means, and killed millions to cause their next enemy to surrender without fighting. They called this "division" where all surrending civilian populations were divided by the number of Mongol executioners present that one day. And they did this routinely. Nothing like this ever happened before in recorded history, or ever since. Not at that scale and for that duration (centuries). For perspective, the Mongols leased Russia to the conquered Russians for longer than the US has existed.

Britain, at no point in its entire history from Roman withdrawal to today, has ever been a global hegemon. They were never even dominant or unipolar. Rather continuously struggling with France, Germany, Spain, and at times the Dutch; nevermind England struggling to first conquer and then rule their own immediate neighbors in the British Isles (Ireland, Wales, Scotland etc). Looking globally, it is true that at times Britain had economic superiority but these eras are measured in decades or at best generations and never approach hegemony or unipolarity on any one continent nevermind all. At or near the zenith of their power, they lose repeatedly. Sometimes to mere colonial farmers. If you pick any random decade for the last 500 years, we can find a significant military struggle for Britain ongoing. Many victories yes. But many loses too, and constant struggle. That is not hegemony.

The fact is the species has been at war with itself continuously for millennia. That was surely happening before we had kingdoms and nation states, too, when the violence had other names. But it was still war.

There is no oversimplifiable theory of war. It persists.

Even the USA and other nuclear powers are obviously still today in a persistent state of mutual deterrence. While the many wars since 1945 have been horrible by any definition, the fact is they pale in comparison to the mass lethality of WW1 and WW2 and we should all be thankful for the relative peace dividend created by nuclear deterrence.

We experienced COVID as ghastly and it was, killing perhaps 1% of all humans infected? WW2 killed 6%. That could have kept happening every generation since. The reason it didn't is #1 the US prevented that from happening, globally, at considerable expenses; and #2 restrained itself, generally speaking, from occupying other countries. Fighting, yes. Conquering and keeping, no.

In this regard, agree with you the US has the best record. For all its struggles and imperfections and systemic issues, nobody has ever done better.

But for the US, we would all be slaves of somebody right now. Either the "Third Reich" or Imperial Japan, or Russian and/or Chinese autocracy pretending to be communism. Even the patriots born there were, slaves to obviously wrong ideas and obviously evil governments. This is easily proven by trying to count the tens of millions of corpses they created. Try. The numbers get over 100M just in the last century, and the unknowable "rounding" errors and margin of error of the estimates is itself horrific and evil. When we're not sure whether Russia or China killed 45 million or 62 million of its own citizens by policy, this is when you should just stop and rethink saying nonsense like the "US's record is shockingly poor." Compared to what?