The flail hold by the Iron Flail is similar to the nine section whip more, which used by Manchu banner cavalries in the late Ming and early Qing period, record by Chinese folks. Anyway, no matter the “flail” you recognized or the “chain whip” I stated, they all are called “鞭” (“whip”) in Chinese as a sort of traditional Northern Chinese weapon instead of a particular weapon. The page of 武經總要 you posted also classed it as 鞭棍 (“Whips and Cudgels”).
Yes, I understand that in Chinese it was called “whip” or “whip cudgel”, but in English such a weapon is called a flail, not a whip. To avoid confusion I translated it as flail and not whip.
Well, the only similarity between the nine-section whip and the medieval East Asian flail was that they were all chained weapons, but the similarity ends there. And AFAIK the nine-section flail was never a military weapon, it was a weapon for self-defense and for martial arts performances. It’s impossible to wield the nine-section whip on horseback, contrary to the two-section flail. If you believe that it was a military weapon, then please provide evidence to support your claim.
All Chinese historical records about 鐵浮屠(“Tiefutu”) I studied show that they were heavy armored elite cavalry troop who was responsible for charging at the front, just like the crusader knight of the West.
Well, yes, but I’ve also seen at least one record which stated that they were heavy siege infantry. Anyways, we already have the Boyars, the Paladins, and the Cataphracts, and if we make the Tiefutu according to your suggestion then it would simply be a repetition of those units.
BTW, cavalry archer + ignoring armor seem over powerful
Can be curtailed in several ways, such as making their speed slower, reload time longer, or more expensive to train.
The Song army developed the cavalry crossbow by themselves. It was not their major weapon actually, however it still had been used by Song cavalries in the battle.
Apart from the painting that I mentioned, there are no other records of their cavalry using crossbows.
In the other hand, the elite heavy cavalry troop 鐵林軍 (“The Iron Forest army”) had been recorded by the Chinese that they used the crossbow. Even if there are few records for the equipment of the Liao army, people believe that they bought the crossbow from Song.
《宋史·太宗紀一》：「己亥，幸新城，觀鐵林軍人射強弩。」
The passage you quoted only says that they shot crossbows, but doesn’t say they did it from horseback. It could very well be infantry crossbows, which were far more common.