That might change depending on difficulty settings, but F1+F3 horse archers kill everything without losses 80-90% of the time when not outnumbered. If you actually command them, repositioning, manually ordering salvos and managing ammunition (F4), they can take anything even double their size (limited only by the amount of ammunition). And that's not all. If you're willing to "cheese" by fleeing once your run out of ammunition or even once enemy melee cavalry gets near (which is the epitome of "cheese" but I've done that in desperate moments) only to "reset" starting position and ammunition without resetting casualties, you can take on any number/type of enemies (except maybe - maybe - horse archers) without a single ****ing casualty.
It's pretty safe to say that horse archers are like panzers (combat vehicles). They're able to hurt while staying safe at distance. The "rock/paper/scissors" element in Bannerlord, if it's even there, is not too prevailing. Horse archers are the best units in open field battles (where they can actually put their horses to use).
I am just wondering, have you guys really played HA in 1.56 or 1.57B? Why do I feel like you are talking about the HA like EA just came out yesterday?
First of all, when not outnumbered, no player can ever get defeated (as long as your troop quality is only a bit lower than you enemy).
Secondly, F1+F3 HA killing without losses 80%~90% is a like a myth. HA battle AI now requires more distance to circle enemy formation and would randomly charge into enemy line if unwatched, so if you have HA only, F1 + F3 is death sentence to 10~30% of your HA units, not mentioning enemy Cav could distract your HA units, making them more likely to run into inf formation. It's really more of a mobile archer that can get shot into pieces by enemy archers, assuming they have some amounts of high tier archers like yours, (and this is not to mention high tier archer is easier and cheaper to get). It used to be that T2 HA can carry the entire battle, not these days. I tried, I know.
Thirdly, I don't see why you can't do the "cheesy" stuff with archer/crossbow unit. I can divide 100% a/c army (preferably Aserai archer) into a few control groups and just play hit and run, and wait till my quiver emptied and restarted again. It probably can do the same, just without maneuverability, and with much trouble because you have to micro everything. HA provides an easier way to micro, so what's the problem. HA rides horse, they are supposed to do that.
Lastly, I disagreed with Rock-Paper-Scissors elements. Battle is won by units functioning and working together. Inf. is a good meat shield to soak up arrows, archers offers a good firepower, cav works nicely as a counter balance to other horse unit and is super powerful when charge from behind, and horse archer is good at harassing and also offers a good firepower. You really just want them all, at least I do. The best of all, battle can be decided by the engagement of individual combat where the effectiveness of the weapon and skills of using it matters. There's no such thing as infantry must be victorious against cav, who said so, Total War? Bow and Horse are both merely an instrument that offers you advantage against your opponents, just like armour and shield, and Bannerlord's combat system illustrated this reality perfectly. Unless you say, Khuzait are dumb enough to only equip their elite HA with just some weak ass arrow when they can obviously offer better. That'd be ridiculous.
The fact is, due to how organic Bannerlord combat system is built, it is just impossible to do Rock-Paper-Scissors thing, because Horse and Bow is just naturally a better stuff to have.