It’s a complex issue. The fact is that NPS campgrounds are operated by a combination of federal employees and private concessions. I’ve camped several where my reservation was made through a private company website, and where I had to check in a concessionaire desk. The rates were purely set by the federal government, and subject to standard discounts for seniors and disabled visitors.
Even when a campground was nominally operated by NPS with a badged park ranger checking me in and uniformed federal maintenance crews, there was a private campground host there to notify of any issues. I’ve also camped at Forest Service campgrounds where checkin was through a private management company that might have also operated a campground store. That being said, I don’t consider those to be privatized per se. My reservation was made through recreation.gov like I’d reserve many NPS campsites.
I think the biggest campground in California is Camp Richardson near Lake Tahoe. It’s technically owned by the Forest Service, but operated by the same family that built it before it was purchased by the USDA. I’ve thought of staying there, but they do everything the old fashioned way - by phone. They have cabins and a campground that are described on their website, but there’s no real time inventory available. I don’t know if the Forest Service would ever consider directly running the place, but I wish they would at least open up online rese4vations.