After further deliberations I have to retract my previous statement. Carvana is actually not half bad.
First of all, you are not limited to whatever selection your local used auto dealer happens to carry.
Second, as mentioned by
@jinga nation, you can avoid the usual hassle of having to deal with car salesmen.
Third, I didn't know that before but you get a 7 day no-questions-asked return policy with a full refund. Much better than the usual 10 minutes with the salesman on the passenger seat trying to upsell you on some warranty or service package.
Fourth, after searching and comparing for days I realized that their prices were actually fairly competitive if you're looking for a car in the 2-3 years and less than 50k miles category as those can rarely ever be found from private sellers (I realize that those are not particularly mustachian, but good luck arguing with a pregnant wife who refuses to endanger her unborn child in a small fuel-efficient Yaris surrounded by massive pickup trucks and full-size SUVs... GO TEXAS!).
Long story short: last night I ended up purchasing an almost fully loaded low-mileage 2017 Kia Soul for slightly over 14k. The next best offer would have been at least 1.5k more, and the average price for that particular vehicle (2015-2017 model) around Austin is around 2.5k more (oddly enough, for some reason unbeknownst to me used cars are more expensive in Austin than they are in San Antonio, Houston, or Dallas).