In my opinion, if they really cared about kids that were less well off, they would provide them with credits or vouchers to let parents decide what is best for their kids.
This sounds like the exact problem I would hope to avoid. Further incentivizing those who can pay out-of-pocket for their children's schooling would presumably create an even more lopsided public schooling system for those who cannot afford any additional expenditure. It may also decrease incentives to improve public schools in those areas. In the US, there are public school districts that can compete in quality with the most desirable private schools, so it's reasonable to assume that there's room for improvement elsewhere.
Also, your comment makes an assumption that caring about kids means allowing parents to decide what is best for their kids. I'm sure this can be true, but it seems unfounded in something as abstract as schooling methodology.
Poverty is the main driver here. Not some "school choice" nonsense.