Tamed animals also can generally reproduce in captivity, with exceptions being exceptions more than the rule. For instance zebras have yet to be domesticated but can be tamed, reproduce in captivity, and we've even created all sorts of hybrids since zebras also can reproduce with horses, donkeys and so on.
GP's point is that for tamed animals, each animal has to undergo the long and arduous taming process before it becomes docile enough to be "useful". With domesticated animals, this isn't really the case - just being raised around humans is sufficient.
Well yes, but it becomes easier and easier over time. And taming over generations can gradually trend towards domestication. For instance Russian experiments in domestication took something like 40 generations to create sustainably domesticated foxes, and that was starting with the cream of the crop in terms of sentiment - they were selectively pulling them from fox fur farms. For another example the horses we know of today are certainly very different creatures than the animals from which they were initially selectively bred.
So I think we're now getting back to the point. The idea that this all had a meaningful, let alone critical, impact on the overall evolution and competence of civilizations just seems quite irrational without some sort of major missing link that nobody seems to be able to provide.
The idea isn't that it has impact on "overall competence" of the civilization so much so that it has effect on their economy. Which matters when it comes to fielding armies, and thus to who conquers whom in the end.
FWIW I don't know if I buy this particular argument from Diamond myself in a sense that all those animals aren't possible to domesticate in principle. What is undeniable, though, is that horses and oxen are very efficient as beasts of burden. Which means that a single farmer can produce more surplus food to feed people doing other things (like say going to foreign lands to conquer them). And militarily, horses give you cavalry, of course, but perhaps even more importantly, they make military logistics that much more effective - and for pre-modern armies the logistics is often what defines their limits.
You're certainly making some true statements, but I don't see how this can be retrofitted to explain the past in a meaningful fashion. So, for instance, Cortes near single handedly (in terms of foreign forces), conquered the Aztec Empire with an "army" of 508 Spaniards and 16 horses. It's not like there was some massive global logistics chain keeping one army supplied. He just had such an absurd technological and political advantage that he was able to convince the locals he was a god, and was able to dominate the local populations by leveraging that - whether in gathering disposable "allies", or conquering more hostile groups. Imagine, vice versa, that 500 Natives landed in Spain. It's quite improbable that they would have found themselves conquering Spain anytime soon.
And food, in general, is not particularly difficult to produce at scale. During the voyage his crew would have eaten nothing more than what the Aztecs would have had available - salted meats, dried carbs, and light alcohol. His primary advantage came from technology - metal working, gunpowder, weapons development, and so on. And all of these technologies were fully available to everybody in most of every part of the world, yet they failed to discover them. And that's ultimately what decided the winners and the losers in history.
So it seems to me that his argument must be boil down to horses causally lead to gunpowder and metal works. And one can try to argue such, but it's quite clearly contradicted not only by the obviously rather tenuous logic there, but also by the endless examples of civilizations which had one yet not the other, in both directions.
Part of the reason why he had a massive technological advantage, though, is that he came from a society that had so many resources to spend on things that are strictly about waging war on its neighbors more effectively. And because its neighbors were also like that - they also had resources to spend on both the tech and the soldiers - both Spain and its opponents were engaged in a brutal never-ending arms race. So when Cortes came to the Americas, he came bearing all the fruit of that. Which in this case was technology, primarily.
With respect to food in general, it's not that it's difficult to produce at scale. But your ability to produce food is inherently limited by three factors - the land, the people to work that land, and the tech those people use. Now, pretty much any agricultural society can produce enough food to feed all the people who produce the food (and all other basic necessities besides). Everything past that point is surplus, which can then be spend to feed people who are not producing food. Which is first and foremost the rulers and the priests - and thus you start getting social stratification - but then also artisans (who make tools) and soldiers (who go and conquer more land to farm and subjugate more people to farm it). And, at some point, a portion of those elites - who have enough calories and enough leisure time to waste on "frivolous" activities not having to do with immediate survival - uses that time to do research that eventually translates to better tools. Furthermore, if their society is in a constant state of war with neighbors - as was the case in Europe for most of its history - those tools are likely to be better weapons specifically.
Horses are clearly not the single definitive factor here, but I don't see why it couldn't be a significant one. This whole setup I just described is clearly a positive feedback loop, so even relatively minor factors introduced early can compound massively over time. And if horses meant that a single European farmer could produce, say, as much food as two Incan farmers, that's a lot more resources that can be spent on waging war and on figuring out how to do it more efficiently.
I should note that the above is not a rehash of Diamond, but rather my own thoughts on this matter. Speaking for myself, I think that it's really the non-stop warfare in Europe, where no single entity managed to unify the entire continent and make it stick for long enough, that was the defining factor in pushing Europe ahead in military science specifically (and other things more or less incidental to that). And then its relatively small size meant that much of this aggressive potential was directed outside of the continent - as military tech progresses, wars of conquest against peer-level neighboring states become less and less lucrative, because you have to spend considerably more resources to conquer the same amount of land and population. Much easier to take all that tech you already have and go curb-stomp some civilization that doesn't have it yet.