>> But while the upsides of specialization are unclear, there are few doubts about the downsides.
What doubts? Winning. To be the best, and being the best is all that matters these days, you must focus on only one sport. For me it was swimming. To compete at the national level you have to train all year, 20+ hours a week. And to actually win at the national level you have to sub-specialize into particular events, specific stroke/distance combinations. Sure, some stars can win at everything, but the majority of swimmers focus. There aren't many champion 50m sprinters who can also medal in 400/800 IMs, not in the last decade or two.
So perhaps the NBA should stop taking only the best basketball players. Perhaps they should look at people who are not literally 'the best' and look at people who are well-rounded and therefor less likely to be injured in their first pro year. But that doesn't win championships. Professional players are essentially actors on a stage for an audience. They are tools for teams, businesses that will be around long after any individual players are injured/retired. What turns the most profit (winning) isn't necessarily what is best for the human players.