Pike squares were amazing but let's not forget the endless amount of drill that was needed to command on of these things; you have 100 men, with 3/4 of the men walking either sideways or backwards. A very novel solution that required a hell of a lot of discipline but the Germans came up with the caracole - mercenary horseman riding up, just out of pike reach, and firing their pistols and riding away to reload. Pretty similar to your horse-archer tactic but with a bit of shock action thrown in.
I don't necessarily think horse archers were a dominant force though; the European standing armies from about the 16th century onwards rarely raised such regiments (Germans and Genoese had mounted crossbowmen!) but they sure did stick around for a long long time. Soldiers of Napoleon's Grand Army were attacked by mounted archers in his Egypt campaign in 1798 and again by the Cossacks in 1812-13. I know for a fact that China still used them but employed far more crossbowmen to counter their swarming tactics. India always seems to prove to be the exception rather than the rule - they are a unique result of strong almost autonomous self governments and mogul states with a lot of European influence. I think, to be perfectly honest, the mounted archer was an excellent raiders unless you pitch them against static defenses - which works out alright until you meet the Poles which don't care much for odds.
It is true that pike squares required a lot of drilling, but so did gunpowder infantry and knights and horse archers had a lifetime of training so that isn't really a difference
Europe is a terrible place for raising large amount of horses, so it is no surprise that most armies there throughout history were based either on infantry or a very small core of heavy cavalry. India is actually an excellent example. European influence was in fact very small there, until the English colonization of India in the 19th century (before that, the English trading posts in India were actually vassals of the Mughal emperors!). Gunpowder weapons were adopted (and improved), but used in a very different way than in Europe, more for sniping instead of shooting into massed blocks of other gunmen.
The Qing dynasty in China which I think you are referring to (ruled from 1644 to 1912) were actually Manchus, a steppe nomad people who conquered China with horse archer armies. They used many Chinese troops to keep their
huge empire under control, but they never fully abandoned their steppe roots which had formed the basis of their success. I think that, like many Euro-centric historians, you seriously underestimate the Asian nomads. From the Saka conquest of Greek Afghanistan and India, to the destruction wrought by the Huns and the Parthians on the Roman empire, and the Mongols and the Turks of Timur Leng (also known as Tamerlane in the English speaking world) on pretty much all of Eurasia, to the Zunghar Khanate, which was the last real nomadic steppe empire (fell to the Qing dynasty in 1759) there should be little doubt that horse archers remained
the most powerful force until not even 300 years ago.