Safeway seems to be run pretty well, after being taken private by PE in the 80s.
Warren Buffet runs Berkshire Hathaway which is acts as a PE firm. Here's a list of everything they own and have owned for decades: https://finmasters.com/berkshire-hathaway-subsidiaries/#cons...
There are plenty of other examples.
PE is just "ownership not through the public stock markets" which is like...the default, actually.