I have no idea how rich you'd have to be to do that. Like upper 1% I guess. I'm a software engineer and the only car I could afford to buy with cash would be like a 10 years old compact. Or maybe a new Dacia if I saved for a year.
That's what I said though. After saving for a year all I could afford would be like a 10 year old car. That's why I'm saying that buying a new car for cash must be the reserve of someone super rich.
It's basically a 10-year plan, but you start by saving for an ~10 year old car for a few thousand and try to pay cash. Any potential car payment can now go into a fund towards your next car, which could just be a newer used car.
If you have a small savings and are a decent driver you can drop collision insurance (which would be required if you had a loan) and save an extra hundred or two each month.
At some point you should have an emergency fund with 3-6 mo of living expenses. A brand new Honda Fit starts at $16k. That's about a yearly salary of someone making ($7/hr) minimum wage working full time. So that should be in the realm of possibility for someone making more than minimum wage (or in a state with a higher minimum wage).
How expensive is your insurance?? I have a fully comp insurance with 20 million euro liability that's about ~$500 a year. If I dropped that down to a 3rd party liability only it would save me almost nothing.
I haven't needed to look in years, but I'm pretty confident it was around that back then. This site [1] says the average is $596 for collision and $192 for comprehensive. So for a brand new car for someone in their 20s I don't think it would be too far out of line for what I would have paid--but wouldn't represent everyone. Comparing to your number I'm surprised how similar the numbers are to Europe.