Can you name any industry that hasn't been given an exemption that hasn't been insanely regulated except sports? Democrats and Republicans, and normal people, supported an exemption for years due to the fact it wasn't offensive to anyone and people liked their team/super bowl/nba final/whatever. But many sports leagues globally operate without a monopoly, including soccer in the US so to say it can't or shouldn't is ridiculous.
Those soccer leagues absolutely operate with a monopoly in individual countries. They even have limits on the number of foreign-born players they can have on their rosters -- which is something you'd never see here in the U.S.
Some sports (like soccer) are more complicated than others because they operate in multiple leagues around the world and are subject to the laws of different countries. In cases like that, the individual leagues end up signing agreements with each other so their player contracts are recognized in every league and the players can't just move from one team to another in the middle of a season.