No it doesn't. It could easily result in less social utility by promoting self-segregation and the concentration of funding into high achievement schools where the marginal utility of each additional dollar is substantially diminished at the outset.
The reason why kids from poor families do poorly in school is also the reason why those families are _least_ likely to be selective in their choice of schools. These are not independent phenomena.
So unless the only utility function that matters is freedom to choose your school, approaches like vouchers are unlikely to improve things and could easily not only perpetuate but exacerbate achievement gaps and derivative social issues, like crime.
The evidence indicates that voucher systems improve test scores, graduation rates, and other aspects of both public and private schools, not just the latter:
Scaled-up voucher programs like those previously advocated
by Secretary DeVos show the worst effects. There have
recently been four statewide voucher programs: Florida,
Indiana, Louisiana, and Ohio. The Florida study is
inconclusive, and the others show large negative effects. In
some respects, the Louisiana results are more convincing
because the results have been corroborated by two different
sets of researchers and students were assigned to vouchers
by lottery—the most rigorous way to evaluate vouchers. In
terms of providing convincing results, the Indiana and
Ohio programs, are somewhere in between, but these show
negative results as well.
Voucher supporters argue that the results have been worse in
the recent statewide programs because they have been
“heavily regulated,” by which they mean the requirements
that students be tested, that these results be made publicly
available, and that schools must let in any student who is
eligible for the voucher. The fact that this fairly minimal
oversight is considered controversial or heavy-handed,
however, only reinforces that private schools are designed
to be exclusive and have little interest in external
accountability.
I wouldn't expect charter schools to actually change much because the evidence is very clear that the most important factor in achievement, overwhelmingly, is the child's home environment.