The thing is, private sales take more work. You have to meet in a trusted spot, inspect the car or take it to a local mechanic, then get a cashiers check or something to pay the person and get the title, then go to the dmv.
For ease, I recommend going to carmax. You will at least know the out the door price immediately. Also they will help you if you need financing.
Yes you will pay slightly more. But it cuts out a lot of the above steps.
Also, I think going below $10k for a car doesn’t get yiu some of the latest safety features. Here’s a good list of highly ranked used cars.
https://www.iihs.org/ratings/safe-vehicles-for-teens