My entire counterargument is that most historical people who fought (and defeated) nomadic cavalry didn't rely on crossbows paired with strong fortifications. We have a wealth of history (more than two invasions) to look towards to see that Chinese states, not only the Song, with inferior cavalry forces performed poorly relative to those states that could maintain strong cavalry forces, in the face of serious steppe confederations.
But the argument is actually about
the Mongols and their way of waging war specifically, not
all cultures who ever happened to use nomadic cavalry. Because those unnamed cultures and the Mongols may well be different in the ways they waged war other than both having nomadic cavalry. Out of the people who actually fought and defeated
the Mongols and won, you have listed two examples, which is good, except that neither of them are relevant to the argument.
Your first example was the Muscovites (assuming you're talking about Kulikovo). They defeated the Mongols in a battle where the Mongols were fighting on foot totally contrary to their usual manner, so that example isn't relevant to this argument. (Faulty logic: "S is better than F, therefore: C must not be better than L against M.")
Your second example was the Mamlukes (assuming you're talking about Ain Jalut). As they defeated the Mongols in a battle with similar tactics
and similar numbers they are not a good example of
comparative effectiveness of tactics. (Faulty logic: "M1 is better than M2, therefore: C must not be better than L against M.")
My examples of Poland and Hungary are good examples because they did worse when they tried to use light infantry and light cavalry against standard Mongol field tactics, and they did far, far, far better after adopting crossbowmen and heavy cavalry and stone castles, in otherwise similar circumstances. And you even have eyewitness accounts that crossbows were particularly effective. Therefore, this makes them a very good example of
comparative effectiveness of tactics against the Mongols. (Solid logic: "L is worse than M, C is better than M, therefore: C must be better against M than L is.")
Those were not an exceptionally strong counter because of operational limitations, not tactical flaws. That's my position.
What you meant by this isn't exactly clear to me, if you could re-phrase it that would be great.
As for why I use whole conquests and campaigns as my examples: because is standard methodology in the field? Because when discussing things broadly, you necessarily take a broad look at things. Especially when it comes to the interplay between weapons and outcomes, looking at events in isolation can produce a distorted understanding because almost every battle, movement or siege is shenanigans on one level or another. So the usual way is to look at trends.
Sorry but I totally reject that it is "standard methodology in the field" to outright
ignore all other potential variables in an empire falling, and without any evidence decide that a single certain variable was the reason they fell, compared to the many other possible variables.
I agree looking at a single example can lead to distortion. But you're victim to distortion yourself because your argument boils down to correlation equaling causation. "Because the Song had crossbows and fortifications, and they lost an empire to the Mongols, therefore the crossbows must be faulty/ineffective." But you have no control example to compare the Song to in order to demonstrate that it was the tactics that were the reason, and not, say, the Song's MASSIVE INCOMPETENCE.
Hungary/Poland vs. the Mongols is a great example because the former and latter invasions are highly similar, with a main difference being more crossbows, stone fortifications, and heavy cavalry being introduced. Therefore, you can safely draw conclusions because you have a control example. You can't do that with the Song. If you were to provide multiple battle examples of crossbows and stone fortifications being the reason the Song lost, then they would count as evidence for your argument. Until then, they do not. They are an example of an empire falling due to poor morale, military inexperience, and severely ingrained bureaucratic inefficiency.
This case is exceptional in almost every way. Xiangyang was carefully designed to be unsiegeable by 1268. It had massively thick walls, reinforced with clay and thick shrouds, with enough supplies gathered within to hold out for literal years. It was surrounded by a wide moat (more than 100 meters), ringed by mountainous terrain and secured on one flank by the Han River.
As an aside, I didn't mention the Persian siege engineers at Xiangyang in 1273 because the investment would have worked without them; the city was written off by 1272.
Sure, the Mongols would have prevailed
eventually, but that's critically missing the point. The question is
how good the Mongols were at besieging stone fortifications, not if they were capable of doing it at all. The conclusion we can draw from Xiangyang is that they were not good if it took them
years of trying to knock down walls that Ismail and Al al-Din could reduce in
days. Emperor Duzong decided to write off the city
because he heard of the Persians' new siege engines reducing Fangcheng, and even then the defenders of Xiangyang still continued to resist
until the Persians' new siege engines began reducing the previously-untouched wall turrets of the city itself. There is no way anyone can argue the introduction of the Persians wasn't a major hastening factor.
But as for why this is a distortion? Because it wasn't the first time it had happened: Xiangyang also fell to the Mongols in 1236, in a matter of weeks.
They surrendered without resistance. Like a huge number of other Song settlements due to the inexperience of their commanders, and overall poor morale in the empire. See, this is why the Song are a terrible example of their tactics supposedly being the issue. Can't execute tactics if your generals either A: don't want to fight the enemy, B: don't have the experience to carry out the tactics, or C: outright JOIN the enemy.
Which Han conquest are you referring to?
205BC-200BC.