My experience looking for a used car in the greater phoenix area says you can't find anything decent for under $10k. Carmax is said to have higher quality used vehicles, but they rarely have anything under $10k (after gov't fees and taxes). Also, in this price range, the mileage is usually 100k+. Nobody keeps maintenance records, so to avoid blowing up your engine, you'll need to immediately spend $500-900 on a new timing belt. Next, expect your high mileage car to have at least one critical part wear out every other month or so.
I searched long and hard and settled on a $6k car. Twelve years old, 150k miles. It was the cheapest car I could find that didn't have any obvious problems (Firestone vetted it; turned out to have $3k+ of problems over the next fifteen months). My income was ~45k. What with all the other bills, I needed a loan.
It's uncommon for a programmer, but the gov't census for the area says a combined income of $40-60k is normal. I know my fair share of single mothers who wait tables for a lot less than half of that.
I heartily dispute the assertion that it's easy to just pay cash for a working car. One last piece of anecdotal evidence to my argument: the ease of selling my high mileage piece of shit car. I put it on craigslist for $1200 and it sold within hours. I described it as clearly as I could: "as is. doesn't work. died and won't turn back on. lots of recent expensive repairs. must tow.".
tl;dr, in my experience, an easy sum to save gets you a car that doesn't run. if you want a car, you need a loan.