Buying used cars is a huge pita. It's also going to be a big time commitment. I've gone through this process twice with teens and limited budgets and I'll offer some observations - not sure these will be of much help.
It's a car for a teenager. Keep that in mind. Teenagers are inexperienced drivers. More than likely there will be at least one minor incident.
I know you want reliability, but in my eyes, of mind, but it's going to add thousands to the price tag. going to a dealer to get certified pre-owned will give you a bit of peace For me, on a lower cost used car for a kid, that's a hard pass. Get Triple A.
Have your kid figure out what car they want. My eldest is short, so she wanted a smaller, "cute" car. She settled on a hatchback, so we went hatchback shopping. In my eyes, the way to get the best deal would be through a private party. Keep in mind that some people become VERY attached to their cars, so when you see this, avoid if possible. We test drove a number of vehicles, and she finally found one she really wanted. I talked to the guy and said, we are still looking around but maybe we'll get back in touch with you. He asked me how soon and I knew I was good. Turns out, his wife badly wanted this car out of their driveway because they just bought a new car and only had a single car garage/driveway and no street parking. Got a great deal on this one, and luckily the car was super reliable for my daughter. If you know a mechanic, ask how much they would charge to bring them with you, cause even if it's just 50-100 bucks, if you have a mechanic give the car the once over, I bet you'll recoup that cost and then some on the purchase price.
Cargurus is the best search engine. HOWEVER, at least by me, this site has been inundated by used car lots. These are fly by night companies (usually) that buy dozens or hundreds of cars at auction, clean them up, mark them up and put them on a lot and on a website. Cargurus has made this process easier for them to aggregate lots of listings. These lots are VERY shady, and don't think for a second that their "mechanics" have done a bit of work on them other than maybe change a hub cap, or rear view mirror. They are difficult to haggle with as this is their livelihood and profession. And they add tons of hidden fees if you are not paying attention. I actually bought a car off one of these lots against my better judgment for my son. I did get a decent price for a car with low miles. There were only a few issues, headlights were burned out which I didn't realize because I drove and bought the car during the day - that was an easy fix, and a tire gauge was faulty. I changed out the headlamps for my son, and brought the vehicle to a mechanic to give it the once over/oil change/tune up. Car hasn't had any issues for 15,000 or so miles, so hopefully I got a decent vehicle.
It's becoming harder and harder to get private sale listings on lower cost cars (at least in my experience). Dealerships aren't a great alternative. I went to Car Max and told the salesperson my daughters budget, and they were like oh no we don't have any cars here for less than $10,000 - have you thought about what your monthly payments could be? I left the lot immediately cause I don't play like that. I don't know if they were lying or not, but I wasn't going to give that person any business.