New schools Chancellor Richard Carranza had quite the week, first signaling his desire to dump the single-test admissions criteria at the city’s eight specialized high schools and then going totally off the rails by arguing that the city’s screened schools “are antithetical to what I think we all want for our kids.”

Carranza is right to want more “opportunity” — but the barrier isn’t in admissions standards but in a system that denies too many lower-income children that opportunity in the lower grades.

And the solution isn’t to tinker with admissions rules to produce the “right” ethnic mix at high-performing schools. All that will do is admit kids into institutions where they’re not prepared to do well — forcing those schools to dumb-down their curriculum, or track the unprepared kids into easier classes (meaning segregation within the school) . . . or let many of them flunk.

Simply ending all screening helps no one: It would just destroy these schools’ ability to remain good — forcing an exodus from the public-school system and sending many families out of the city altogether.

On the specialized high schools, Carranza doesn’t even have the power to change the rules for three: Stuyvesant, Bronx Science and Brooklyn Tech’s admissions are governed by state law.

A chancellor can mickey with the rules for the five other schools — but the results may not be what he wants unless he imposes outright quotas: A 2015 NYU study found that most ideas for broadening the admissions criteria would do little to improve racial diversity. In fact, some alternatives would reduce the share of black students admitted.

But maybe quotas are what he has in mind, since he’s now condemned the rules that allow other high schools and middle schools to accept only kids who meet certain academic standards.

City Councilwoman Inez Barron (D-Brooklyn) voiced the radical version of that complaint: “Every institution in this country has embedded racist policies in it. Bar none.”

Well, let’s be clear about who’ll be the big losers under a quota system: Asians, who are now the largest group at most elite high schools yet plainly don’t benefit from any embedded racism.

Fine, if anyone can find evidence of middle-school administrators using screening to unjustly keep minority kids out of a neighborhood, get them fired. But that’s not the systemic problem.

The simple truth, as we’ve said before, is that the main public-school system is actually two different things: About a third of the schools do a decent job (or better), while the other two-thirds just go through the motions.

Parents with the resources (whether it’s money or a community that strongly supports education) move to areas zoned for decent primary schools, and the kids move up through the “good” system — learning enough to qualify for a good middle and high school. And children condemned to the failing schools rarely catch up.

That’s why more than two-thirds of students flunk state tests and less than half of high school seniors graduate ready for college or the work world. And why increasing numbers of black and Hispanic parents opt their kids out of their zoned schools, into charters or out-of-district schools.

The problem isn’t money: The city spends more than $24,000 a year per pupil in the name of educating all these kids, of every color — indeed, it spends more on its bad schools. It’s the resistance to the kinds of radical reform that might offer real opportunity to the “lost” children.

If Carranza can’t manage to increase the number of good city schools, anything else he does will be shuffling deck chairs on the Titanic at best — and, at worst, steering straight into the iceberg.