But they obviously know that unless they cheat they won't win, so the desire to cheat is predominant in their minds, thus proving that it is a form of mental illness - QED?
Cheating is defined as violating the sanctioned rules. We(society and all its representative bodies) institute rules so we may predict the behavior of others. Its comforting to assume everyone subscribes to the same rules, but this is simply not realistic. Whether the prize is a gold cup, the admiration and envy of others, or a coupon for a large pizza, each of decides how much effort is warranted to win.
Rules are arbitrary and often include clauses which make weaker contestants more competitive. Lance Armstrong is a famous cheater, but he peddled his bicycle every meter of the race. He didn't take a shortcut and meet them at the finish line. It turns out, he's a cheater. I can understand why a grown man with no marketable skills would think winning a bike race was important, especially after returning from a contest with cancer, where there are no rules and sometimes a doctor will remove a testical, or some other body part.
Lance wanted to win and he did. No one can dispute that. He also violated an arbitrary rule that actually had nothing to do with riding a bike. He didn't push anyone over or let the air out of their tires. Maybe he is mentally ill, but mental illness is defined by a set of rules no less arbitrary than the rules of the Tour de France. It's entirely possible to construct a set of rules which defines lack of desire to win at any cost, to be a human failing, and thus a mental illness.