I'm basing those suggestions off my time playing cRPG mod. While a lot of that stuff did apply to the real world in a smaller scenario it's not as effective. A lot of times, at least in that mod, any cavalry that dared try to run archers down just got headshot off their horses. As for shields, half the fun of the game, IMO, is the directional attack/block system, and when you have a one hander+shield your best attacks against anyone was the left-attack because it aims at the head, and the shield acts as an omni-directional block which throws out the directional block mechanic altogether. Sometimes I just wanna reach out and tear that dumb shield away from them or something. I think the only way to really deal with shielders was to have weapons that did bonuses to shields, and those weapons were typically worthless for anything but breaking shields. I suppose couch lancing does tons of damage to shields but it's not very reliable since the cavalry typically didn't last super long (and if they did it's because they're god tier wearing full plate and full plated horse).
M&B combat is still janky and not reaaaally authentic, but it's close enough to still be fun and at least a little believable. A lot of times it's either spin-to-win or people doing all sorts of weird twirly feint attacks that look ridiculous but are effective because they make it hard for the enemy to judge your attacks right.
E: Sprinting. They could add a stamina bar that only applies to sprinting. It would take a long time to 'recharge' so you'd get a good few seconds to charge in, but you wouldn't be able to just sprint all over the place. There'd also be a sort of acceleration period so it's not an instant janky movement.
Re shields: That's kinda the point of shields. To give people a way to bypass shields would be to render them useless. They're designed to be the thing you use to block attacks. Using axes against them helps but again, you're fighting a person who has a big piece of wood and metal specifically made to intercept your attacks (even attacks from the side, that's why they're wide and not bucklers) which means they are at a distinct advantage.
You could make it so that a person with a shield turns around a bit slower but you need to be careful with that because if you slow down the turn rate too much, shields are useless things that aren't worth the effort.
You could make targeting people's exposed appendages easier but given how games work, that'd just mean everyone would be going for leg strikes in MP and instead of shields everywhere, you'd get people leg slicing all over the place (or whatever method of bypassing shields you can think of).
I cannot think up a solution that'd make shields and two-handing a big sword both viable options without losing authenticity of medieval combat (SHIELDS EVERYWHERE) and replacing an overused overpowered strategy with another overused overpowered strategy. (except the aforementioned slowdown of turning, tho that'd need to be carefully handled)
re cavalry charging archers: I did not have that problem playing vanilla SP. I've never played a lot of MP mount and blade (mostly because I feel the game is better in SP) but I suspected that headshots might be a big thing from people playing archers. Just like in other MP games when playing against experienced players and the skill level and the like. However, this issue should be relatively easy to fix just by boosting the level of protection a helmet gives against piercing ranged weapons.
re jankiness: it's true, M&B combat is janky AF. A lot of it has to deal with the game itself (the animations look weird and stunted, tho from what I've seen of M&B 2 it looks like they somewhat remedied that) but I suspect most of that feeling of jank comes from the input method itself. A truly jankless melee combat simulator is something that one can only achieve with VR, I feel, given how integral control of your movement is to fighting in close combat. So there's a fair bit of jank to be expected from any game that tries to simulate it. Again, a lot of that can be remedied with improving presentation and adding polish (better animations, more responsive controls, the like).
re stamina bar: That sounds like a pretty good solution to me. Probably would also be affected by what kind of armor you're wearing and how many weapons you have and your various stats and the like, but it sounds like a decent solution.