Yeah no Warband ,as good as it is shouldn't be considered as a source for this kind of discussion, a well trained pike formation would decimate any kind of heavy cavalry head on, as horses, even well trained ones are not stupid, they know that running into a wall of pikes will kill them, hence the square formations during musket warfare and schiltroms and phalanxes before that. Melee cavalry would be good against a pike formation as they had little to no shields, and if kept well supplied they could even take on well armored and shielded infantry, but if they run out of arrows then they are pretty useless unless they are used to attack flanks, scare enemy into thinking they will be charged, or pursuing fleeing enemies, in this respect Total war is a bit of a better teacher on the uses of cavalry than warband*
Another thing that comes up a lot when reading about the mongols is that they were not ashamed of running away from a fight, and while European and Arab horses may have been faster, their horses had crazy stamina, and once the enemy was thoroughly exhausted they would lay ambushes, so in that way I think that their horses were superior to the more powerful, faster and bigger western horses, they also mostly wore leather or light armor while Europeans were weighed down with mail and other stuff.
*ACTUALLY in war band, with clan matches I remember cavalry having a mainly harassing role during the opening phases of combat, either making infantry move out of formation so that archers could shoot them, or flushing enemy archers out of cover so friendly archers could shoot them, in all of these they would play a passive aggressive role and only once the infantry locked together they would go in and bump people to the ground so infantry could chop them up, also having to constantly check behind your back for cavalry will put you at a massive disadvantage.