Guns, Germs, and Steel presents a very important argument that geography plays a critical role in the course of human social and economic development. To the extent that geography is important, it excludes alternate explanations that may be religious, racist, or nihilistic.
The orientation of the continents - North-South for the Western hemisphere and Sub-Saharan Africa, and East-West for Eurasia and the Mediterranean - helps dictate the extent to which people, animals, and plants will move. A culture can migrate from Iberia to China (or from the Levant towards both directions) while staying reasonably within the same range of latitude. So plants and animals that are adapted to one area can more reasonably move to another.
If a climate is stable, then the domesticated plants and animals (wheat, cattle, etc.) can be domesticated in one area and moved to others. And if the domesticated stock dies out in an area, it can be replenished by a bordering area. The resulting gains to food production eventually multiply, and more people means more food producers and thus more domesticators. More domestication and food production continues the feedback loop.
As the society grows, that leaves more people free to do things other than eat. Religion, politics, art, carpentry, metallurgy, etc. Because of geography, if it is easier to travel and trade, then non-agricultural gains (to society or technology) made by neighbors will travel more easily.
Of course, the population growth also contributes to crowd diseases and agricultural diseases. A lot of Eurasian plagues and diseases came from domesticated animals (not necessarily direct contact, but it could be from waste, or consumption, or from fleas/mosquitoes - all of which require close coexistence). So the Eurasians were constantly experiencing diseases and many survivors retained antibodies for these diseases.
So the result is that the Aztecs and Incas had reasonably developed societies and sophisticated systems, but they had far fewer domesticated plants and animals (fewer nutritional options). They were laboring within limited geographic spaces such that there were fewer candidates for domestication and fewer avenues for interaction. What candidates there were had trouble moving between North, Central, and South America. Less interaction and smaller societies means less of all the gains made by Eurasians, who won the food-geography lottery. So the Western hemisphere was less well prepared, had fewer people and less developed technology.
Most critically, it lacked the antibodies to Eurasian diseases. Taking their fleas and rats with them to the New World, the deadliest part of the Columbian Exchange was the disease. This may have killed millions of natives, probably more devastating than all the horrible intentional acts committed by the Europeans. This may have been the most powerful benefit to the continental axis of Eurasia.
The Genetic Rebuttal
Your argument that the environment selected for traits seems unlikely. First, because humans are so crazily interrelated that the variations tend to be relatively small even over a few tens of thousands of years. Second, because Europeans were actually counter-selected for survival in the Western Hemisphere or Africa, with many of them having pigment designed for low-sunlight climes. Third, which genes made Europeans better at building ships or guns?
The genetic argument tends to be closely akin to the religious argument in actual application. It's largely non-falsifiable unless you can point to specific genetic traits. So in practice both the religious and genetic argument follow the form of "X favored this group of people so they prevailed," where X can be God or genetics.
Occam's Razor: simplest answer, ceteris paribus, tends to be right. In that case, it's simpler to say that the continental axis allowed movement of more domesticable species, which led to agriculture, which led to more people freed from producing food who could then eventually do things like build large ships or do metallurgy. Combine their crowd diseases from agriculture and the ships from their spare population, and they had the means to travel West and let their diseases and weapons wreak some havoc.
Throat-Clearing and Other Thoughts
Obviously, there's no excusing the horrible fate of the indigenous peoples of the Western hemisphere. Actually, the fact that it was mostly an accident of geography should be taken as a strong argument AGAINST justifying the depopulation of several continents of people.
In my opinion, Diamond somewhat de-emphasizes the role of indigenous decision-makers post-Columbian exchange. Almost all the footholds gained by Europeans in the Western hemisphere came when locals decided to ally with the strangers for gain against their domestic opponents (Massasoit and the Wampanoag, Aztec opponents, Incan counter-claimants, etc.). That said, the technical question is why the Spanish sailed to Mexico and killed Moctezuma rather than the Aztecs sailing to Rome and killing the Pope. For that specific question, the indigenous politics are not so relevant (but they are relevant to the implied question of why the Europeans managed to populate the other hemisphere).
Personally, I happen to buy more into the Rational Optimist, by Matt Ridley. He is also an evolutionary scientist, like Diamond, but his views is more human-centric. Humans have the ability to trade, and trade produces surplus benefits. So humans are best served by being interconnected in larger and larger groups. That's more or less my own view, which is somewhat distinct from Diamond. I only mention it because I suspect people interested in Guns, Germs, and Steel might enjoy reading the Rational Optimist.