I don't believe the NBA or NFL necessarily have the world's best athletes. Just because they have the tallest and largest doesn't mean they have the best. Certainly some of those athletes are excellent, but we in the U.S. have no reason to believe Americans have the market cornered on athletic ability. Before my time many claimed that Jim Thorpe was the greatest athlete ever, and some people still say that QB Sammy Baugh was the best all-around football player of all time.
Personally, based on what I've read, what sports I played (baseball, tennis, and football), and the films I've seen, I think nothing is more difficult than hitting a baseball consistently for both average and power, without steroid enhancement, that is. Consequently, I'd call Ty Cobb, Rogers Hornsby, and Ted Williams athletically about as good as it gets. Williams also throws in fighter pilot skill, by the way, along with fishing skill perhaps the equal of anyone who's ever fished in America. But I've seen soccer players turn in astonishing performances too. I saw little Diego Maradona do things with his feet and a soccer ball that defied credibility, especially in the World Cup.
Ultimately, if pressed, I might have to fall back on the King of Sweden's statement a century ago that the winner of the Olympic Decathlon may be the greatest athlete in the world, as the Decathlete is no specialist but has to master ten different track and field events to emerge as the winner. Perhaps Bob Mathias, who won the Olympic Decathlon in both 1948 and 1952, is our greatest athlete ever. He not only did it twice, he did it without the artificial stimulation of supplements and PEDS. I don't watch the Olympics any more - too much commercialism, glitz, and cheating - but I was mesmerized by the strength, conditioning, and all-around performance of an American teacher in 1968 by the name of Bill Toomey. I know that the night Toomey, running in the dark in Mexico City in his final event, won Decathlon gold I felt sure that he was the greatest athlete I'd ever seen. And, like Bob Mathias, Toomey did it the old-fashioned way - he earned it.