Safeway might be OK for shoppers, but it's terrible for the workers. My Grandma worked there from 1965 to 1985, and watched the decline in the 80s. It was a really good paying job with a good union and good benefits when she started. My brother and cousin worked there a decade ago. Today, the union is a joke, and the hours and pay are the same as any other minimum wage job.
Berkshire Hathaway is a public company and IIRC Buffet only owns about 15% of it. So it really that different from a large corporation with many subsidiaries which also runs an actively managed investment fund at the same time?
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.