Density of infantry in any given point of battle will be higher than density of cavalry. So, if 1000 Huns are in this particular area of battle there will be 2000-3000 infantry, so not one pilum but 2-3 will be tossed at every horse and then riders stabbed with swords. Heavy cataparacts might be tricky if they are really good. But such heavy cavalry is always very expensive, so how many of them could be there in the battlefield? I think, that late Roman army weaponry reflected late Roman soldiers. Variety of mercenaries were used and they fought with weapons that they were comfortable with.............. so.
One Pilum, 1,000's of enemies, you'll come into contact with a minimum of 10 different enemies.
Stab rider on the ground lol, you have to get him off his horse first while he's slashing down on your or spearing at you from up on a horse.
What about a Persian Cataphract, armed with a Kontos lance, try getting him off a horse while he ploughes right through you and even when he's mired down your gladius still can't really reach.
No disrespect but you described a very "Action movie" type scenario, realistically its not that easy and one Pilum isn't going to help you defeat multiple opponents.