So I just finished a course on the History of Warfare so here's my take.

It's important to first understand that most of the killing on a battlefield was done during the retreat, something light cavalry is excellent at. It is significantly easier to break an army and send them into retreat by breaking their formation. In the case of steppe tribes, to do this against settled people they employed hit and run tactics and encirclement with horse archers. You would ride close, fire off some arrows, pull back, rinse and repeat. People don't particularly like to just sit there and be pin cushions, so they would venture out, especially when that nomadic army would go into a feigned retreat which was one of the favorite tactics of steppe tribes.

By going into a feigned retreat, it would trick the opponent into thinking they'd won, and hoping to get involved in the slaughter, they would break formation and chase down the enemy. And when they did that, the steppe warriors would turn around run roughshod over the disorganized masses chasing them. It even worked against heavily armored knights, as the chargers they rode were bred for power not stamina so the steppe warriors would just withdraw until they ran out of steam, then kill the Knights who were now basically sitting ducks. This was the tactics from the Scythians to the Magyars to the Kipchaks to the Mongols etc. By the time of the Mongols, steppe tribes had started incorporating both medium and heavy cavalry to further give them an advantage in tactical flexibility.

To your question about infantry. One thing the Mongols were great at was choosing their battlefield. You would harass and harass until the enemy became fed up and would fight on conditions favorable to Mongols, meaning a wide open battlefield with room to maneuver. Infantry are shock troops; if you don't engage them in hand to hand, they can't exactly kill you, so the Mongols would just fill them full of arrows until they broke or fell for a feigned retreat. The horse archer was the premier military system until the introduction of firearms because of the mobile firepower it provided. It was so dominant it led to settled people's like the Byzantines creating an empire wide training program for cataphracts, which were heavily armored horse archers, to combat them. The Mongols were just the culmination of centuries of steppe horse archers tactics.

They were defeated sometimes, and a good example of a conventional European army defeating a horse archer army is Richard the Lionheart's campaign in the Third Crusade. To avoid encirclement, he marched his army along coastal roads so the sea was always at his back. When faces with feigned retreats, he kept his knights in reserve and did not send them chasing out. Instead, he lined infantry in the front and had ample crossbowmen and archers behind to return fire. Eventually Saladdin's army became impatient, got too close, and Richard sent his knights to charge them. Crucially, they stopped and came back after breaking the horse archers, avoiding tiring out their horses.

Source: Erik Hildinger "Warriors of the Steppe"

Just to add to what's already said:

It's important to remember that such rock-paper-scissors that you find in video games don't necessarily match reality. This is especially true in pre-modern times when few armies were professional and had significant training, and battles often came down to morale. For agricultural societies, the limited count of horsemen often have more to do with cost and supplies than tactics. For if you can bring horses, why not bring them anyway. If you run into a situation where horses are unsuitable for the situation, you can always dismount and fight on foot.

The Mongols (and knights and samurai) prefer to fight mounted, but it does not mean they can't fight dismounted. And this is the reason that I personally like to distinguish the "cavalry" of classical and early modern period with the "horsemen" warriors who just happen to have horses and know how to fight mounted. Case in point, in one battle, half of the Mongols dismounted and fought on foot. And they still broke through the enemy lines.

Finally, at this point, foot soldiers of agricultural societies were often hastily assembled conscripts/levies, with little to no armor, making them easy picking for Mongol archers (foot or mounted). Their lack of experience and training also make them relatively easy to break through for toughened, experienced Mongol warriors.

See the Yuanshi section for Muqali in Chinese here