This was true about testing too back in March-April, but it would have actually looked bad if the NBA and other leagues did have most of their players and staff mass-vaccinated while many others are struggling to create appointments or aren't even eligible yet. Most of these players are between the ages of 20-40 and aren't eligible in most, if not all states yet (unless they have pre-existing conditions which I also don't think many of them do). So I kind of think the NBA and Silver strategically said this to flame any potential fires.
If you remember back in March 2020, some NBA teams actually took up nearly half of the state's tests available having their players and personnel tested for "precaution" while hospitals barely had tests at the time. It led to a lot of outrage. The big example was the infamous Jazz-Thunder game that was postponed which basically suspended the league, and I think half of Oklahoma's tests that night were used on the Jazz and Thunder players + staff (since test capacity was very limited back then)
And maybe I'm in the minority, but I absolutely hate this nonsense of "oh let the players and rich guys (young politicians + celebrities) get voluntarily vaccinated so that others can believe it works". That's BS because the main problem at the moment is actually distributing the vaccines, and right now a good chunk of the U.S. still can't actually sign up for an appointment since they aren't eligible or some states really screwed up in their vaccine rollouts initially. Some hospital systems have already suspended additional appointments because they don't have enough vaccines now. If you let others just "cut the line" then you're just contributing to the current system in place and are content with it basically.