I guess I should have clarified that motivation is the key issue here; there is none, not for academics, anyway. I know very well that a motivated scholar can overcome a bad K-12 education: I did it myself. That is what is throwing us; we were both strivers, and this lack of academic ambition is just not something we understand very well. (This kid is artsy. Perfect grade in music, for instance. Math or English? Not so much. LOTS of zeroes for missing deadlines, etc.) Also, FTR, the military isn't an option, due to a disqualifying medical issue.
I'll add that the kid is more in favor of private school, mostly to get away from the prevalence of poverty (or what the kid perceives as the prevalence of poverty. The truth is that very few of the students at their current G&T public middle school live in dire poverty; most are the children of civil servants or skilled blue-collar workers, many of whom are single parents. The kiddo is engaging in a fantasy that a private school will magically be devoid of substance abuse, bullying or less-than-nurturing teachers -- a fantasy that comes from recruiting videos rather than reality.)
What I'm trying to get at is a balance point where, hopefully, some part of the family does not unduly suffer for the choices of another part of it. DH & I have already been through spending a ton of money to try to give kids the best options, only to have all the options left discarded because of a lack of motivation. It hurts us quite a bit, both financially and emotionally, and we're trying to decide if it is fair at this point to put our own needs first. Our children are not going to be able to contribute to our living expenses in our old age; even if they were inclined to do so, they don't have that kind of earning power. (Not caring at all about educational quality isn't what we are looking at; we're not that angry.) We're 22 years into the active parenting gig; and while our income is solid, we're not anywhere close to what I would think of as well-off, and amassing a 6-figure college fund required years of concerted effort.
DH & I have friends & colleagues who are very invested in getting their kids into the "right" college, and watched as they stressed over Ivy League interviews and/or lamented the performance of a kid who failed to exceed a 4.0 GPA in high school. Thoughts like that never crossed our minds even before we knew what our kids' educational aptitude was; we don't have those kinds of expectations. (We both were first-generation college graduates who felt privileged to attend our state schools.) In our book, average is just fine.
The high school situation here just doesn't offer that much in terms of middle ground. There are very good public options for high-achievers, but if you don't do well enough to qualify for them by the time you are 13 years old, then what is left is pretty substandard unless you are in a position to invest a lot of money into a private education that is what I would class as mediocre. I have nothing against being average -- what I question is whether or not it is worth investing an estimated $60K into the education of one average student before higher education is even on the table. (Note that that price is for a non-selective private high school; we could spend 2X that on a really good one here if the kid could get in; but if those kinds of grades were on the table then we wouldn't need to spend it at all, because the kid would be high-achieving enough to be accepted at the local magnets, which are just as good, if not better academically. At many of the most select private schools, the investment is as much about the connections as it is the academics, but that's another topic for another day.)
PS: Trust me, homeschooling is NOT an option in this situation. This kid needs the structure of a classroom and the skills of professional educators to get a decent outcome.