Couple of issues here.
Throwing lances (henceforth listed as TL) were the perfect ranged horse counter. Hitting infantry with them is a dice-roll, but a horse coming at you easily fits inside your crosshairs. I seriously think you have some sort of God-complex when you use statements like "Plus they won't stop me from hitting your head." I can't speak for EU throwers, but NA throwers were great counters for cavalry.
Secondly, if you are on a horse, you don't need a side-arm. Side-arms are used in a situation that you are trapped or stuck in beyond your control, and your primary weapon is not effective. Because horses have insanely high speed, they can pick the cream of the crop of situations. Once you are dehorsed, well you probably won't need your lance anyway at that point. Having a lance drop when you switch weapons isn't much of a drawback.
Pikes and throwing lances, however, suffer from this greatly. Pikes cannot pull out a shield to cover themselves from archer fire, and throwing lances are pretty weak choices for melee. Granted, they did get a melee damage improvement, they still wouldn't be my first pick as a weapon. Because you are so slow on foot, you're going to be put in a situation where you have to fight your way out when you would like to run away.