Mongol rule in Russia posed no serious threat to western Europe in terms of imperial conquest. The horde was fragile on its western frontiers. Steppe invasions and conflicts on the east between the Mongols and Chinese empires shaped that area for millenia. Russia, as an outlier, if we consider it a part of Europe, is uniquely exposed to the steppes in a similar way to China. The Mongol threat to greater Europe, however, was not that great. The tactics, ecology, and technologies, that made them a remarkable threat would not have been effective in western Europe during the same time periods.

It's certainly an interesting "could have been", but you need to move very far away from what actually happened to make it a convincing possibility.