My company started out with an entrepreneur/founder, moved on to a private investor (two rich guys) and have been owned by equity owners two times in a row. We're just at the end of the road with the current owners, and it's ... interesting.
Both have been similar. Short term and cynical. They have no interest beyond maximizing profit when they exit.
I'm guessing that Silver Lake are brought on to help expand the business. They'll want to prove a model where they can grow quickly with high earnings in the new clubs/investments. If they can prove that this model works, they will be able to sell their "shares" with a really big profit, within five years.
EBITDA/earnings will be key. And with players being the easiest way to keep consts down, there is a risk that there won't be as many expensive signings the coming ~5 years.
I'm using my own limited experience as reference.