I almost always use cav, even on some rather tight maps. I mainly use Rouncey, Steppe, and Courser, and occasionally use the Destrier or Cataphract. My current build is mainly polearm, with a bit of throwing (which I usually only take some war darts when I play as infantry).
The damage from
melee weapons against cav is fine. I can make it hard for anyone to hit my horse with melee weapons. I have the speed and maneuver to be able to choose my fights, so melee damage is mostly avoidable. I don't need the horse to be a tank that can plow through enemy territory without much risk of being cut down. I'm happy to fight on the edges of the enemy. It just requires a little more self-control, caution.
I haven't had much of a problem versus
throwers either. Throwers can be avoided fairly easily as cav on most maps. You don't need to be too far away before you're out of their accurate ranges. You can just about dodge their throws at 20 feet. So if I see a thrower who sees me as I'm running towards them, I can usually call off the charge and turn away. I'm fast, so I can leave and go find a more advantageous opportunity somewhere else on the battlefield. And I can come back later from a different angle and catch the thrower when he's not looking. Or I can even just prance around in view of the thrower to keep him distracted. You just need patience.
It's mostly only on really tight maps that I often find myself too close to throwers for too long. There is often only 1 way to approach the enemy group and no room in the narrow street or corridor to go past the group, so I'm left either trotting in a circle away from them, or dashing desperately through their line to attack, get to the other side, and make myself a hindrance for them to worry about in the rear. But this is just an issue with the map being unfavorable to cavalry.
Though I will say throwing lances/spears seem unbalanced, since they seem to take out any horse instantly no matter where on the horse they hit (followed by a 1-hit kill while the rider is on the ground). But since people are saying that throwing is getting a bit of a nerf soon, I suppose this might not be an issue for much longer.
Crossbows do a lot of damage to horses, but I'm fine with that. Their slow reload allows me to wait till they fire and then approach them before they can complete the reloading. I avoid charging a loaded crossbow that is aiming at me, since they can take down the horse with one properly placed shot. It's a lot like how I deal with throwers.
Bows are the biggest threat to horses. These are what force me to always be hiding at the beginning of the rounds, waiting for allied infantry to get in close to distract them. I could charge a single archer in the open that is aiming at me and be able to kill them if they don't dodge the attack, don't stun me as I attempt to attack, or aren't able to get a shot into the vital part of the horse. But taking on 2 or more archers in the open who aim at me during my approach is probably going to be the last charge I can complete, if it doesn't get me killed before that.
So basically, my cav strategy revolves around staying away from archer fire. On most maps, I can usually manage to find a place to fight that is blocked from archer view or is far enough from archers to make them miss a lot and not be so deadly. And there is often a way to approach archers quickly from an angle that is out of their view so that I can be close before they notice me. If they only are able to get 1 shot off before I can attack, that's a fair trade.
The biggest problem is on maps with hard-to-access platforms that archers can stand on and get a wide view of the battlefield. Sometimes it is walls/ramparts that overlook the field. Sometimes it's buildings or towers. Sometimes it's just a steep hill that either has no flat access point (for cav to approach from quickly) or is up against the boundaries of the map. These archers can hail down arrows without much worry of being caught out without support.
I don't know if I'd say that archers are unbalanced though. If archer spam is really bad on a particular map, I could pull out the cataphract, lose some money on upkeep, but at least be useful for a while. I can do that since I save money by using cheap weapons (crappy shield and war spear to go with my heavy lance) and a Rouncey on most maps. I can live with that.
But I definitely would say that horses don't need a nerf.