The NBA, MLB, NFL, and MLS have limited exemptions from antitrust law. The MLB has a
complete statutory exemption from antitrust law. (See
https://sportslaw.uslegal.com/antitrust-and-labor-law-issues...) However, their monopoly status doesn't prevent competing leagues from forming; indeed the modern NBA and NFL are the results of the merger of smaller leagues. The MLS itself is the third iteration of professional soccer in the US, the first two having failed quite miserably, and multiple professional football leagues have launched and failed in just the past decade despite healthy ticket sales due to mismanaged spending.
UFC is not a monopoly but does engage in anticompetitive behavior, including for example restricting its fighters from competing for other promotions, controlling the sponsorships they can receive, and even the sponsors they may promote in the ring. Their practices affect their fighters, but have no discernable effect on their audience. You can easily choose to watch one of the other promotions, like Bellator.
WWE is the oldest and most successful wrestling promotion, but it has a dozen or so major competitors, and there are hundreds of wrestling promotions at the local and regional levels.
Gymnastics and the other Olympic sports don't have monopolies, why do you think they do? Track athletes compete in multiple professional events over the course of the year (most of which aren't televised in the US due to lack of audience but many of which are broadcast globally), and most olympic sports, gymnastics included, are not commercial draws outside of national championships and olympic qualifiers.