Cav is useless? Horse archers suck? Wow. Show me any case where your all infantry army is able to beat a full stack without triple digit casualties.

Horse archer/lancer army can easily do that. Foot archers also don't have the tactical flexibility of flanking and crossfire as well as effective rout chasing. If your cavalry can't even get there before your infantry routs the enemy army, then you are fighting a decidedly unequal battle in your favor. Try a closer battle and see how cav superiority can win the day for you.