> PE puts very little of their own money into the deal though, while they own it they don't buy it.

This is simply untrue. A typical PE firm creates a GP fund using LPs money. This GP fund typically does a management buyout (which means 50.1% of more) of several companies using a mix of equity (e.g. GPs capital) and debt (banking lenders). So by every definition of the word, they absolutely own the company.