Cav still gets in done in my opinion, just most horsemen are enormous cowards. Either an armored horse that can take a lot of projectiles, or an Arabian Warhorse, which makes dodging ranged, for me, trivial. This is as both Jav-Cav or 1h Cav, which I think is the best ranged counter.
15/21 with shield skill, and either ATH or IF. Get a Spathion, Arabian Cavalry sword, or similar, and it works great.
Killing archers is really damn easy as cav, as long as the teams are balanced and the map isn't shit. In a good round, the archers will be supporting the infantry blob fight by targeting pikes/other ranged/whatever and cav can sneak in and cut a couple down if they stray too far away from supporting polearm infantry. This is pretty fun gameplay.
In a bad (aka a typical) round, the team will be 1/2 or more ranged who are all blobbed together, camping a hill or some other shit part of a map and creating a deadly crossfire to anyone within shooting distance. I'm sure every infantry player has experienced having to attack something like this at the end of a round, and being decimated by bowstagger, 1 hit KO throwing, xbow headshots, and everything else.
A cav, even heavy cav, get torn up just as quick like that. Focus firing a horse when you need to get in close to get some kills will drop the rider and horse within a couple seconds. A good horseman will avoid being in these situations or wait for infantry to support them and cause distractions.. but more and more there's barely any infantry to fill this role, or the maps are so bad that you can't even reach most of the ranged.
Then the occasional jarid, bodkin, or bolt will just inexplicably black bar even a heavy horse with some speed bonus voodoo from across the map. This is prob the most rage inducing and what makes cav stay the fuck away from ranged most of the time. They do so much damage it doesn't even matter what horse you have.
I usually stick to medium horses (Destrier, War Horse) and just target infantry players with lance or HBS, because its way more fun to fight those guys in the melee blob battle and support your team with bumps. Hell its usually easier to kill an infantryman than a ranged since you can juke pikes, but not the hoard of Agi build archers/throwers that crossfire and randomly sprint around.
At least xbows you have an opening when they go to reload. Throwing and light-bow archers just shoot out missiles, and decent archers can hold their shot infinitely long until you get close and stagger you with a shot. Its dumb.