A couple of things come to mind.

Didn't the mongols ravage poland? That's Europe. Meaning Europe did not get spared.

European philosophy and math started back with ancient greece. They were ahead of the rest of the world in geometry and math in the year 1000 B. C.

But empires come and go. The next empire took the science and further improved. The romans, Macedonians, ottomans etc. Europe is not one people.

At the time of the mongols the Catholics were in charge of most of Europe. Kingdoms were organized. "Backwater" might be a bit of a stretch?

There was war between the Christians and the Muslims at the time so Europeans might have been expecting attack. It might have been very difficult for the mongols to conquer the European territories.

Millions of Europeans preferred USA over europe for the last three centuries because the opportunities were scarce. Europeans were mainly a farmer people so land was equal to prosperity. They gave out farm land for free back then. Which one would you prefer?