"We want sports to be fair and we take elaborate measures to make sure that no one competitor has an advantage over any other."

That is incorrect. If sports were fair, then everyone would win. Sports are specifically not meant to be fair, the entire point of the competition is to reveal which competitor is superior to the others.

The rules that we have against performance enhancing drugs are not about having an advantage over other competitors. We don't have similar rules governing how much weight training a particular athlete may do, or how clean their diet may be. They are about establishing rules by which universal play can be governed. The rules of the game define the game. If you play by a different set of rules, you are playing a different game. If the Tour de France did not have a rule against excessive[1] steroid use, then using steroids would not be cheating. But, the Tour de France with steroids is not the Tour de France, it's a different race entirely.

As a culture, we associate healthful living with physical fitness, and we associate sports accomplishment with physical fitness, therefore we make the huge leap of associating sports accomplishment with healthful living. Even sans-drugs, sports accomplishment doesn't necessarily equal healthful living, especially in the cases of ultra-marathons increasing risk of heart attack and the head trauma of American football increasing the chance of dementia. But we still make that association, so as a culture we tend to value sports that at least pay lip service to maintaining the image of athletes as healthy. Though, it's important to point at that we are inconsistent in that regard: certain performance enhancing drugs are allowed in horse racing, and why shouldn't pain killers be considered performance enhancing, if they allow the athlete to perform better than they would without them?

Our culture shapes the rules we choose for our games, the rules define the game. Cheating is not about "gaining an unfair advantage". It is an unfair advantage for the New York Yankees to have the very best sports doctors in the league, but it is within the rules of the game, therefor it is not cheating. Much of sports strategy is about finding ways to find unconventional play within the rules to give oneself an unfair advantage. Cheating is violating the social contract of the game: "we've come here today to engage in a competition and these are the rules by which we will evaluate the competition".

[1] steroid use rules are often defined in terms of testosterone levels, and the limits are frequently well, well above the levels expected for a healthy male. The rule is thus, "no high testosterone", not actually "no steroids", and it has become standard practice in many sports to try to cycle a steroid usage to peak testosterone levels just below the limit. This takes significant personal experimentation and blood testing to see exactly how the individual body reacts to the drugs and over what time frame. But don't expect the difficulty to mean that people aren't doing it: they are. When you see an athlete testing positive for steroids, it is not a question whether or not they were using steroids or that others were also using it, it's a matter of them not following their regimen properly.