They can and were trained to do it, but no commander wanted to sacrifice his cavalry in an ill-timed charge (meaning a charge against disciplined infantry not otherwise engaged). Against peasants in the feudal age? Sure. But they were not the main brunt of an assault.
Their main usefulness was in their mobility, getting around to flanks or behind to archers. There was a lot of study and tactic changes to counter that kind of assault, because at least in western europe cavalry weren't going to just charge into an enemy formation like in these games. You had infantry for that, and cavalry to hit the flanks in a pincer move. Battles were all about positioning to try to limit that cavalry assault as much as possible while allowing your own.
I have two very good books on medieval combat somewhere and I practice mounted combat myself every month. I don't remember too many details but I do remember the historians involved in writing these books did say cavalry wasn't used the way a lot of people think, nor was the dominance of cavalry quite so pronounced except for a smaller window of time earlier on in the medieval period.
It depends. it also depends on the specific period we're talking about. The time of Bannerlord with Eastern Roman cataphracts definitely had head-on assaults. Even at Hastings, you had head-on cavalry charges with the not-so heavily armored horses. At Carrhae, Parthian cataphracts attempted to crush the Roman lines. At Vienna, Jan Sobieski's hussars completely blew apart the Ottoman lines. The go-to move for the French at Agincourt was a (wildly unsuccessful) cavalry charge.
Much of the point of these heavy cavalry attacks right at the front is to completely ruin the cohesion and formation, with the infantry not far behind to engage the shattered formations, as opposed to getting into a prolonged melee with the enemy infantry. Or, they think they can make them run for it and then kill them as they rout.
A cohesive line is difficult to break with infantry and archers alone. It would likely take a tremendous effort and a lot of time for infantry to successfully break a large front of the infantry, even if they had a significant advantage. Rows of determined men can be difficult to simply rout with other normal men. Thus comes the point of heavy cavalry. If they are well-trained horses, well-trained horsemen, and well armored, and enough in number, it is very, very plausible that they can smash the enemy's front and effectively end the uncertainty of battle right there.
It would be easier to coordinate that than to pin down the infantry with your infantry and have the cavalry flank all the way around to the back. This wasn't very commonly achieved. Alexander the Great was a master of this tactic, and that's why he was so good; this hammer and anvil approach is difficult to pull off, and he knew how to do it. With most other commanders and armies, often the cavalry at the flanks would either be bested by opposing cavalry or forces, win and chase down the enemy cavalry going off elsewhere, or ending up doing something else entirely, like attacking supply trains, as opposed to attacking into that big scary area of people, as they are likely to still have protection for the flanks. It's not like the army commander can radio them, the cavalry is usually on their own at the point. People like to do the less dangerous options when it's their show now. Usually, decisive cavalry attacks to the flanks were well set up and well-coordinated far before the actual battle begins.
Most cavalry, (specifically light cavalry) throughout history were for, as you say, scouting, skirmishing, engaging enemy cavalry, dealing with archers, flanks, etc. This was also the bread and butter for cavalry after gunpowder ruined the usage of heavy cavalry. In a fair few periods in history, you had heavily armored cavalry that was highly trained and completely capable of crushing infantry formations from the front as well. But you are correct that the primary role of cavalry throughout history wasn't to smash open infantry lines. However, I do not think this was nearly as uncommon as you make it out to be.
