The power of horse archery is not in stats or gear or anything like that. It is in the idea of being uncatchable and ranged at the same time. Enormous power lies in the fact that (as most cav) you can choose your battles. You can choose, when, who and how to engage the enemy. From high position you also have great overview of the battlefield. A wise HA chooses that lone unaware player. Either he cuts is head off with a scimitar at speed, or he harass him with arrows. A wise HA kites players away from their main group.. Takes enemy cav for a spin to the other side of the map etc.
From an armoured horse, very little except other cav - which takes a huge risk chasing you - can touch you. On a quick horse, they will have problems reaching you..
Alone in a 1 vs 1 situation, the only counter to HA albeit temporarily is throwing. Archers will never get the chance to draw another arrow if the HA combines bumping and shooting, let alone an xbowman will never be able to recharge. Throwing forces the HA to keep some distance, but since their accuracy is much greater than thrown weapons, they win that one too.
Now.. The only really dangerous HA out there are the ones with 3-6 times loomed bows and arrows, or sometimes horses too. Mostly only veterans have 6-9 loomed items so its not a big deal, but it might be eventually.
And yes, 1 HA may disrupt, but not destroy a flanking run by ninjas, but 2 is often murder. 2 HA working together is simply deadly, and much much worse than meeting 1 after another.