They are streaming rights, they are paying for eyeballs. The exclusivity is why there is competitive bidding.
If the league let three other services match Apple's price, then it devalues what apple pays for by (at least) 2/3rds.
The leagues don't have to give exclusive streaming rights, but they should be prepared for the offers to drop by well over half, as well as a lot of interest to drop off.
What makes Apple different is that they want worldwide exclusive rights. For some parties (like MLS), thats huge because it gives them a huge boost in worldwide coverage - and they weren't going to get comparable bids in other markets anyway.
CarPlay isn't an exclusivity play though. Most receivers support a native OS, as well as CarPlay and Android Auto. CarPlay is basically an App in the car's native OS - it basically turns the car's console into an external screen to the phone. It is purposely designed to be pretty easy to support on the car side, because they can't dictate significant CPU/storage requirements and aren't coding bespoke apps onto manufacturer's platforms.
It is an indicator in the "doesn't care what his customers want" column at most.
Hypothetically if Tesla cared even more than they do today about what their customers wanted, they'd share their good and valid reason for not supporting CarPlay.