The awnser why the Romans defeated the Gauls is the same awnser for why the Romans did not the defeat the Germans, basicaly leadership.
Both Caesar and Arminius where significantly smarter and/or in better control of their forces than their opponents were and thus they won.
Of course, the better training and the use of superior weaponry, armour and existing tactical level formations are also factors, but beeing Bigger helps.
For me, it is extremly diffcult to elavuate who had the better tactical formations, who had the better weaponry/armour (I know that Chinese top class weapons where better than their roman counterparts, but how many of their troopers acutally had them?).
Objectivly elevating things like Logistics is also extremly difficult, even more cranky is the diplomatical/politcal situation. Around 0 BC, Up to 40% of Romes force where busy in Germania. I do not know how "annoying" the Viet/Xiognu whatever tribes on the Chinese borders where, but they would certainly be a factor in an actual war.
Every single one of these points would propably have a bigger impact than height, but, at least to me, every single one of these factors would be even more uncertain.
So I hope that everyone can agree that The Size of a Warrior has a positive effect in a mellee battle.
Lets get to the second point, showing or indicating that the Romans (not really the Romans, I was talking about their auxillarys when I brought the matter up) are propably bigger than the Han Chinese.
Lets start with a general essay how the size of someone is defined.
The Genes define the Size of a given person. Size is an extreme example for Polygeny, literal hundres of Genes are directly involved, propably around 10.000 indirectly (f.e. if you have a genetic nutrition difciency, you size can be modified because you need more to eat)
They can be modfied by Mutation or, more likely, recombination with an other Genome. The Genes do not say how big said individual person will be, they say how big he will be if he has enough to eat.
The correlation of eating and Size is not a mutation but a modification, this means that although the person looks different, his Genome stay the same.
This means, a genetically big person who is small because of Malnutrition could give birth to a kid, that, given decent nutrition, would easily outgrow him. Because this can occur on a very sudden basis and for a broad spectrum of persons, modification are usually easily appearant and do, in general, not have a big effect on the "perceived" size of an entire nation.
This is the most basic fact.
However, this model would not explain why people all over the world are getting bigger.
Here our good friend Darwin comes into play.
Because of facts like beeing Stronger, sometimes faster and more attractive to the other gender(the propably most important factor), bigger persons usually enjoy the benefits of natural selection compared to smaller ones. Given enough food, the Big Guys will reproduce more rapidly, leading to more "Big Guy" Genomes leading to more Big Guys and so on.
The speed of this varies with the nutrition. Nothing to eat does favour smaller persons, however, a very long period of "nothing to eat" (1) has to happen to have any traceable effect on the Genome of a Bigger population.
Therefor, the average size usually goes up if there is enough to eat, or stops/declines slowly if that is not the case.
As very long nothing to eat periods are a) rare and b) occured in both Europe and China, we could propably await a similiar speed of "growing".
This would mean that todays height differences would propably also be present (though propably in a slightly different quantity) in the past.
Although China had some times of extreme poverty, so did Europe (30 Years war beeing one the few examples for the afromentioned nothing to eat period).
As it is, apart from mass Migration and a really big number of inter ethnic marriages(which is a strong point though, the (assumable small) Mongols never settled in Europe like they did in China), I see no reason why the growth curve for China should be any different from the growth curve for Europe.
Assuming a similiar development of the Sizes, Scenarios in which Han-Time Europeans would be bigger than Han-Time Chinese would be indicated.
Just to sum up my Theses:
1: Size matters in a combat engagement as, of course, do the armament, the available formations the training and a bunch of other factors.
I feel that a discussion about this topic has a place in a military history forum.
2: Han-Time Chinese were propably smaller than Han-Time Roman Auxillarys.
This is based on the fact that this is the case today, in addition to the point that size changes usually happen at the same speed (outlined above).
Because I feel that detailed discussions about population Genetics do not belong to a Forum named Chinese Military history, I would ask persons who disagree with me of the second Thesis to PM me.