The Mongol conquest was a lot more complicated than that. Russia would have been a non-issue anyways. The areas that they really exceeded all reasonable expectations anyways was China and Iraq. In both cases however, they only had to destroy a few armies--Not a small task, but certainly attainable.
The Mongol conquest of Iraq was far easier than you make it out to be. Whilst Iraq required a major campaign and was by no means an easy conquest, its fate was pretty much sealed with the defeat of the Kharezmians under Jalal ad-Din Mingburnu.
In fact, I'd even go so far as to say that one great archievement in western Asia in which the Mongols exceeded all reasonable expectations was the defeat and destruction of Khwarezm (which, thanks to Jalal ad-Din Mingburnu's almost ASB-ish luck, took no less than two major campaigns), which was not only the most powerful polity in west Asia at the time, but also had a massive army of horse archers that were on par with the Mongols in therms of skill, weaponry and fighting prowess.
After Mingburnu's defeat, the Mongols had (relatively) little trouble conquering northern Iran, the Caucasus and Anatolia, giving them a good base of operations against the Abbasid Caliphate and the other remaining Muslim states in the region.
The only reasons that the Mamluks won was because of the length of each supply chain, Mamluks were steppe warriors themselves--at least tactically, and that they had superior numbers. Which isn't to say that the Mongols could have destroyed the Mamluks, logistics rules in the Mamluk Sultanates favor.
Whilst logistics and geography certainly did prevent the Mongols, and later the Il-Khanate, from invading Egypt, the Mongols failed to avenge the defeat at Ain Jalut because of the war that had broken out between the Il-Khanate and the Golden Horde.
Berke Khan of the Golden Horde even went so far as to ally himself with the Mamluks, and it is the war with the Golden Horde that prevented the Il-Khanate from organizing a proper campaign against the Mamluks, which in its turn allowed the Mamluks to establish themselves in Syria and the Levant.
Also, the Mamluks' numbers were
never superior to those of the Mongols or the Il-Khanate - even at the battle of Ain Jalut, the Mamluk army was pretty much the same size as the Mongol army, and take in account that the bulk of the Mongol army had temporarily returned to Mongolia along with Hulegu Khan at that point.
The Mamluks didn't exactly use the same tactics either - they were more specialized in close combat (IIRC they defeated the Mongols at Ain Jalut because they managed to corner the Mongols between them and a cliff), whereas the Mongols emphasised archery more.
But even if the Mamluks would have used identical tactics, it still wouldn't have made much of a difference, though - the Mongols had already faced, and destroyed, much larger armies of experienced and highly trained horse archers in Transoxiana, northern Iran, Anatolia and Iraq.
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Nonetheless, your main point still stands though - the logistics indeed did favour the Mamluks, and even when they were operating from the Mughan Plain, the Mongols were still at a disadvantage in the Levant.