Yes there are private lodges that charge those amounts, you are not reading wrong, and I doubt you will do much cheaper booking elsewhere though I stand to be corrected on that.
However there are many many incredible lodges that charge much less.
It seems you are looking at the lodges in the most ''elite'' catergory, presumably in the Sabi Sand, where we have stayed several times ourselves but have never paid anything even close to that. We sill get luxury and an amazing experience but do not choose to include massages, top shelf liquors, spa treatments, designer linen and so on.
Check out all the lodge reviews on this site and you will get an idea why they are so expensive. All inclusive rates, service of top quality rangers and trackers for anything from 7 -8 hours upwards per day, usually great food at the lodge and also out on the drives, amazing rooms and suites, and of course very limited vehicles at a sighting. From our experience the rates are no less via an agent, but thats not to say its not possible, you may do better.
We too cannot begin to imagine paying $3000 per night but I can assure you plenty of people do, But I repeat you are looking at the very top lodges for that sort of money, no need unless you want all those aforementioned type of extras.