And especially, note that unions are wage fixing cartels!
Besides that though there are a few other things that don't come up in white collar environments. One is career tenure. I think NFL is the shortest average career length for North American sports at something like 2-3 years. In MLB non-elite players who are 30+ are now becoming less likely to get picked up by teams after hitting free agency, meaning the median career for a successful career is something more like 6-7. Athletes in such a situation don't have any leverage to sit out a whole year, which can represent 14%-50% of their lifetime earnings in the primary thing they spent their youth training on. Athletes also have to worry about physical injury that don't exist in white collar environments, which has led, for example, to changing touchback rules in football and adding a 26th man to the roster in baseball this season.
You're burying the lede there: the careers, particularly in sports like football, are short because sports actively break people. Athletes can tear their ACLs and alter their ability to walk or run for years, or end up with traumatic brain injuries that permanently alter their personality. After 5 years they're used up and cast aside -- and who in the public remembers a 2nd string Linebacker?
Given the immense amount of training, suffering, and injury... I'd come to the same conclusion re: bargaining.
Most European countries have a professional football players union, and their contracts are owned by the club, and not within a monopoly like with American sports. The same goes for referees, coaches, managers, etc.
Again, the amount of money these people are paid can be very high, and while many players don't use the union for things like negotiating pay, they rely on the union for other types of representation.