Private schools are self-selecting. they are also size-limiting, and when times are good, have competition for spots so that underperforming (in any way) students are selected out.

your local public school is required, by federal law, to educate to the best of their ability any student in their catchment area. If that child needs 24 hour care in a boarding program for special needs at $150,000/year (which happens) they get it. If they need a tutor to sit with them full time in the classroom, they get it. If they need English as a Second Language instruction, they get it. If they need remedial math, they get it. (well, all of that is theoretical, of course, they are supposed to get it) anyone who uses that many resources in a private school will be sent away, for the most part.

Also, $13,000 is nothing in the DC education market. let's take the relevant private schools in DC and look at their tuition (not their spending per student, which is usually 50% higher, just tuition):

St. Alban's (boys): $26,501. for day students.
National Cathedral School (girls): $24,724
Capital Hill Day School (elementary only): $20,520
Edmund Burke School (high school, co-ed): $24,950
Georgetown Day (k-12 co-ed): $24,303
Sidwell Friends (prek-12 co-ed): $25,990


hmm, maybe the Catholic Schools are better? (remember, St. Alban's and National Cathedral are Presbytarian, attached to the Cathedral)

Georgetown Visitation (girls, 9-12): $18,350
Gonzaga College (boys, 9-12): $13,000

so even at the good Catholic schools, heavily subsidized by the Diocese, charges at least the amount spent on each student in DCPS simply as tuition.

and, full disclosure, my beloved alma mater (the Catlin Gabel School in Portland, a much cheaper city: $20,900.