Does it make sense that a mounted archer have higher damage output than a foot archer, let alone a crossbowman?
No. Horse archers used weaker bows than foot archers. Or at least the strength of the shot was a lot weaker. You don’t have your legs grounded to anchor your weight, so your top draw weight is lesser. Even if you were using the same bow, you were not able to overdraw as you would standing up, simply due to the constriction of your hips. If you wanted to shoot on a horseback, you couldn’t turn your hips to get the full draw length to begin with. Shorter draw length = less power. A foot archer can stand 180 degrees Olympic-style, getting the full draw of his arm, perhaps even more. Good luck drawing a monster 200lbs warbow sitting down, let alone achieve full draw weight. Mounted archery also takes great deal of training.
WIth these in mind, maybe the devs will have a better direction on how to go about balancing mangudai (too weak), camel archer (too weak), and horse archer (too strong).
Mounted archers should be relatively expensive and not expendable, hard to mass up. They should be units that are able to get high return-on-investment if micro’d correctly. Really bad ROI if you suck at the micro (noting the high level of training required).
They should have high hp, light-armored, and as fast as scouts (with the exception of camels). Mangudai and rus horse archers should be infinitely kitable against melee units. No melee cav should be able to catch up to them.
At the same time, decrease their range and missile damage overall to match regular archers (noting the weaker power of horse bows). Perhaps keep the high rate of fire (also noting the high training received by mounted archers). Missile infantry should easily counter them by out-DPSing them if massed value-for-value. There is absolutely no sense in keeping mobile missile units as a front-line fighting force.
The Rus horse archer meta forces you to make archers (which do softly counter horse archers), but they are not hard counters by a long shot. Since horse archers don’t cost that much, a Rus player can get a mango or two. A single mango shot can tip the scale.
I sometimes think that unit balance might turn out alright if historical facts were relied on, trying to reflect truths about military units onto be game practically.