So what I see happening is that the public schools that fail as I said will either be forced to find a better way to educate their students or shut down.
My main concerns with this line of thinking are the following:
What does it mean for a school, public or private, to truly be failing?
How can you force a school to provide adequate instruction if it does not have the resources to begin with?
Again, I do not expect you or anyone to have an answer because these are quite complicated problems. My fear would be that, without fully understanding the above concerns, we will repeat making the same mistakes, no matter the economic model through which education is provided.
For example, a school in a very poor district will have much lower standardized scores and graduation rates than the average; no matter the resources it is given. Perhaps it may appear that a selective charter school in such a district outperforms regular public schools. But we have to remember that selective schools require that its students' parents have enough awareness to submit applications to it for their children. It would be reasonable to assume that such students have better home environments, on average, than others which will enable better performance in school. So it is unfair to compare a selective charter school with a regular public school that is charged with instructing all pupils, regardless of how proactive their parents are in pushing education on their children.
Many public schools are clearly failing to provide adequate instruction to their students. While we can see such outcomes with our eyes, do we understand the reasons for why? We need to study these issues if we are going to actually fix the problems.
I'm wary of notions of "increased competition", which rely on the assumption that markets are always efficient. This is decidedly not true for public goods, such as mass education, where the profit motive yields a mismatch of incentives between beneficiaries (entire society) and education providers. The full benefits of mass education are not seen until children are in their 30s. On the other hand, schools in a competitive system need to meet short-term metrics (standardized tests, graduation rates, etc.) that can be and are gamed for improving the school's standing.
This is not a simple problem.