One has to look at it from a cultural aspect, as well as population size, and how much the country publicly or privately contributes to their athletes.
Look at the three countries you stated, Russia, China and the USA. All three are of the largest countries in the world. In terms of land mass and population.
Culturally, those are very athletically inclined societies. They take things like combat, fitness and sports very seriously. Not necessarily collectively, or the USA and Russia wouldn't be so fat and drunk, and China would treat it's people a lot better than they do. But in terms of focusing a good deal of attention on those aspects of society where it counts, which to them, is in what they compete in their own respective countries, as well as what they will compete on the world stage. All three do not leave much room for disgrace, or weakness when one commits.
And look what the results are. (Not trying to be racist) but Asians and people of Asian descent are usually good at sports like gymnastics, lifting, judo, Karate and other sports that utilize reflexes and quickness such as table tennis and things of that nature. Caucasians (Russians and most U.S. athletes) tend to be good at sports like swimming, high jump, distance running, lifting, field games (shot, javelin) wrestling and sports that require focus like archery and etc. Africans (U.S. athletes included) tend to be good at sprinting, long jump, middle distance, various field games, boxing and sports that require quick bursts of power.
If you look at the sports these countries tend to medal in, it's pretty evenly dispersed according to that and their respective populations. Culturally, those countries and all of their sub cultures even have traditionally encouraged the sports they tend to dominate in.
U.S. Swimming is the best in the world. Better than Europe, better than Australia (either of which, the majority of other good swimmers come from). I heard in many countries, people learn to swim at a very young age. Many U.S. swimmers however, start even younger. Like, the first year or two of their lives. Which I know sounds like an extreme. I'm not saying Michael Phelps did that or anything. But a lot of people do. Same with China and gymnastics. And Russia in various track and field events.
It's pretty strange how it is similar on a world stage. Just look at the funding. It's almost exactly how they function as a society. China funds it's team with state capitalism. Russia uses a mix of government and private, with more of an emphasis on government. And the U.S. athletes train primarily in the private sector, with a little bit of government.
Interesting question you pose, about these three world powers.