it seems you are mixing up leveraged buyouts with PE
PE is investing. Limited partners give pool some money, the manager is the general partner, they also put some money into the pool, to align incentives, and that's it.
Everything else is not PE itself, but whatever strategy the fund has, or whatever deal some traders have came up.
PE is finalisation of business, its ownership is far more similar to a mortgagee than an owner in every sense of the word.