Is it fair to conclude that the vast majority of Europe being sparred by the wrath of the Mongols, while formerly more advanced and wealthier societies in the Arab world, Persia and East Asia were utterly destroyed, led to the former's eventual rapid rise to global dominance? I know there's other factors often cited for this, like the mobilization of armies in the Reconquista (and the eventual influence of many of those soldiers post Reconquista in exploration), post-plague peasant revolts in Western Europe leading to the relative demise of serfdom (and thus the unleashing of more human capital), the competitive forces between hundreds of kingdoms, states, cities etc... helping to spur technological development, and the sacking of Constantinople and the transfer of ancient knowledge out of that city and into present day Italian cities ... Among many other factors. However, after reading about the utter destruction to these societies by the Mongols, and the "luck" that most of Europe was sparred, appears to me to really the biggest factor.
Sorry for any typos. Typing on mobile.
OK, there is a glaring problem with this, in that the Mongols were not responsible for the utter destruction of any civilisations. I keep seeing this crop up amongst Redditors, and "Mongols" has pretty much become a go-to answer for questions in /r/history on why the Islamic Golden Age ended or why the Middle East did not develop as swiftly as Europe and things like that.
The Mongols were responsible for brutal, localised destruction, but the long term effects of this are hotly debated. Most Mongolists these days tend to stress the fact that the Mongols contributed an awful lot more to world civilisation than they took away.
To begin with, there's the fact that Mongol destruction tended to be concentrated in certain areas rather than a complete sack of everywhere they touched. The wars with the Jin Dynasty, the war against Khwarezmia and the war with Russia were the places where we can see the real destruction the Mongols are famed for (although, as Morgan notes, the extent of the destruction reported in Kiev doesn't entirely bear out with reports of a flourishing and thriving Kiev a few years later, and it's thus debatable whether Russia really was affected as badly as the other two. But I digress). So the only places to suffer the worst of the Mongol destruction were Transoxania, the Khorasan, parts of Afghanistan and Northern China; and though these were important regions, they did not constitute the entirety- or even the bulk of- Islamic or Chinese civilisation. Furthermore, though there was a lot of destruction, it did not spell any kind of total desolation in these regions. Several cities maintained or improved their prosperity in the coming years.
I haven't mentioned Baghdad, which was also subject to a brutal and notorious sack. But Baghdad's capture- frequently described as the point of total destruction of the Islamic Golden Age- was really not that important. Its library was great, but it was simply the most important of many libraries in the region, meaning that the amount of knowledge lost was not too catastrophic. It became, in later years, a convenient scapegoat for Islamic historiographers to explain the inability to keep up with the West, despite the fact that Islamic societies continued to prosper for many centuries more and that the Islamic Golden Age was either well over or would maintain itself for another two-and-a-half centuries, depending on who you ask. Furthermore, the advances in science and culture made under Mongol governance were extremely great, largely on account of the cultural exchange promoted between the two Toluid courts, but I won't go into that now. Thomas Allsen's "Culture and Conquest in Mongol Eurasia" is the best book to read on that score.
So the trouble is that you cannot look to the Mongols as a direct cause of European prosperity and Asiatic decline- historians have, by and large, moved on from that way of thinking. I believe that the argument has been made that the loss, upon the empire's fall, of the trade routes which the Mongols restored and maintained drove Europeans to find new ways to acquire eastern luxuries and wealth, but I'm really not sure how convincing that is. Either way, there were other, more complex factors at work which caused Europe's rise.
Probably of little to no importance. In the post-Mongol period life generally went on as usual and eventually in virtually all areas the local people and traditions absorbed their Mongol rulers, I don't see why this would be any different for Europe.
A possible impact I can think of is perhaps if rival post-Mongol leaders tore Europe apart in dynastic struggles trying to revive the Empire (in similar fashion to Timur further east) and kept the region unstable and unable to become the colonial powers that they did become. But since the Europeans fought each other endlessly anyways the impact of this is negligible.
I have a few issues with this conclusion, but the biggest is that you are taking a somewhat essentiallist perspective on Europe and other regions, as though it is legitimate to see them as individual units that are inherently separate. But can we really think of "Europe" as a single unit, particularly in discussing its "rise" (what does this mean? Who was rising? France? Albania? Estonia?). Also there is a bit of "Clash of Civilization" perspective here, as though Europe can only rise when other regions have fallen away. But Europe's rise, such as it was, was inextricably linked to the health and security of other regions. After all, most of the silver mined in the New World ended up in China. If the Mongols had actually destroyed China there would be no way that Europe's history would have been as it was.