Some manufacturers only offer 3 years though with no option to extend so the value proposition is not nearly as good IMO if you're buying a vehicle with only a year left on the warranty.
Q: Who drove this car?
A: Oh, whatever random people wanted to pay me money.
Q: What kind of maintenance did you do?
A: The bare minimum to get it to last a year.
Q: What options did you get?
A: The lowest ones the manufacturer would sell us.
Does that appeal?
Also, if you buy a car thats 2 or 3 years old you can actually figure out what is likely to go wrong with it, something you generally don't get with new cars (unless you're buying a model is about to be refreshed)
I generally find the maintenance required to keep the warranty generally exceeds the cost of repairs that might actually happen. It's not exactly rocket science to change air filters, oil, transmission fluid, and brakes.
The other thing about cheap used cars is when they break, you get a new one. I'm not going to spend $2k on a new engine, I'm going to buy a new car for $2000-4000.
Things worked out well for me. The Corolla was a tad scruffy but otherwise presentable, and my mechanic gave it a clean bill of health (except for two of the tires being over-worn, which Hertz replaced). It's been a while, but I remember the price being ~$11k (4dr sedan, 43k miles, about 1.5 yr old), and all said and done came to ~$12k (taxes, titles, fees, what-have-you). The car was young enough and under-mileage enough to still be under manufacturer's warranty. The Hertz warranty was 60 days.
The 3-days rent is not much risk (maybe $200) versus $500 deposit most used car dealers want. This is probably not an easy way to go if you need to do this on a car loan. But if you can raise the cash, I think this is a good way to go.
And many manufacturers offer an extended warranty on used cars. Buy a 1 year old "certified" car many times has a longer warranty than an identical new car at a lower cost.