My understanding is that the MLS has a "strict" salary cap, except it allows big exceptions for people like Beckham.
Overall I think football has managed itself fairly well over the years. About half the clubs who kicked off the football league are in the PL (or were last year) and most still exist.
Where it has gone badly wrong it is due to dodgy owners and more of these are around now football is big business. Portsmouth had a few: the son of an international arms dealer wanted for illegal trafficking, two penniless Arab billionaires (one who might not have existed), and a friend of the arms dealer. Birmingham has the hairdresser with triad connections who suddenly got rich. And who knows what is happening with Malaga. Then you have Bates and all his secretly owned companies who do him lucrative favours despite having "absolutely no connection" with him. The guy at Rangers is another. And finally you have leaches like the Glazers and Hicks-Gillett. Proper vetting of owners, something the NFL do well, is what football needs more than anything, making sure they have money to buy the club and won't dump the purchase cost on the club or sell off the stadium and other assets.
I'm less worried by Sheikhs and oligarchs if they actually put money into football, even if it does upset the competitive playing field. Mansoor and Abramovich have invested nearly £2 billion in football, money paid to footballers, managers, coaches, and other club employees (plus agents). UEFA wants to stop this, while allowing the Glazers to take £500 million out of football, giving it to bankers instead of footballers, managers, etc. It seems backwards.