The ~$15k ones tend to be Catholic or otherwise religious, and quality varies from "worse than the local public schools, actually" up to "notably better than the local public schools, though not as good as a really good private school".
Some cheap secular schools are only a little more expensive than the bottom end of the religious ones, but they don't tend to be much better than average public-school quality (nb. that may still be a lot better than some particular public school district)
... which doesn't necessarily mean preferring a better school that's only better due to selection bias (and knock-on effects from being able to remove very-disruptive or low performing students) is a bad way to go, at the individual level, if one has that option. The observation's more relevant to policy-making, than to the relative merits of those options to any given family.
One of the consequences of our state (Washington) going to a state-wide student funding model is that public schools really don't want to lose their non-special education students, because they get a fixed amount of money per student and if they lose too many of the students to private that don't need (and don't get) extra resources, they don't have enough funding to provide programs to the rest.
Yep, absolutely, and that part of their ability to select their students is a major factor in the "look, private schools can educate kids for less than public schools spend, with similar or better outcomes!" sort of stats that get passed around.