Well its a serious reduction to explain the Mongol empire's success as simply being due to the use of horse archers. Horse archers weren't new, they have their strengths and their weaknesses and they certainly weren't the only reason why the Mongol empire was so successful.

Firstly on the subject of horse archers themselves. Their strength is of course the fact that an army that was almost entirely mounted had a great deal of mobility. Now under certain circumstances this does give them the choice to more selectively choose to engage weaker forces or disengage in unfavorable exchanges but depending on the necessities of a campaign you might not be able to choose not to engage a target under certain circumstances. It might for example be a bad idea to leave a fortified position in your rear. And under these circumstances horse archers aren't known for for example an inherently particularly great ability to go storm a city. Additionally there's limitations on whom can produce horse archers of a considerable quality and quantity. Specifically steppe empires can do because a large potion of their population spend much of their lives riding horses, they have great opportunity to train in horse archery via hunting on horse back and their territories tend to include large regions of graving land which allows them to support a very large number of horses allowing them to equip their people as such. While certainly some sedentary empires have used horse archers, for example the Byzantine who particularly began using them once Mongol influences began to inspire changes in the Byzantine military, for a country that's mainly arable land and urban populations its harder to produce the same kind of numbers of troops who can fight on horse back and have them be as skilled in this style of warfare. But sedentary civilizations do tend to be able to produce greater population densities. So empires like say the Han could raise large numbers of crossbowmen and archers who particularly from field fortifications, the Han for example sometimes used chariots fortifications. These large numbers of crossbows and archers can provide densely packed fire power from cover with supporting spear men being able to form up to ward off any light cavalry charges. This is the kind of tactics that people like the Han used against earlier great steppe empires like the Xiongnu to reasonable success.

So what was the Mongol's secrete? Firstly Temüjin instilled into his warriors a greater degree of military structure and discipline into Mongols under his command. He separated his troops into distinct units with levels of command, units split into smaller units each with a dedicated commander who would be either voted on by their subordinates or at the top levels appointed by Chinggis himself. This made it much easier to both be able to coordinate a plan prior to an engagement then during the fight send orders down the chain of command as the situation unfolded as well as for smaller unit commanders to react to changes in the battlefield at their own level. This made it much easier to reliably carry out those kinds of feigned withdraw tactics that you describe which without these kinds of tools of command and control are pretty difficult to coordinate on the scale of a major battles.

The other major thing was adaptability. It tends to be a strength of the most successful steppe empire that they were able to turn the conquered nations into subjects from which they could draw more troops. But the Mongols were most successful when they did this on a large scale. Being able to adopt things like siege tactics from their enemies and particularly in China raised large armies of from their new subjects to continue their wars.

Its also pretty evident that the Mongol invasions of Europe weren't unstoppable because they were stopped. The golden horde which was one of the successor states of the great Mongol empire lead by Mongols who had invaded Russia. Though they successfully conquered many of the Rus principalities and the like in the long term their attempts to drive further west into Poland, Hungary and Serbia were ultimately repulsed.