There are two arguments that have been brought up.
1. This sort of disjointed system is not desirable as it propagates inequality in opportunity. This should be fairly obvious and not contentious.
2. Private schools and these pods, as their extreme form, face a similar issue as insurance markets if there is capture in governance (which seems to be the case in the US). On the one hand, wealthy people have an incentive to bundle their funding in private schools, which will lead to higher average contributions. They also have an incentive to petition government not to use their other tax contributions for public school, as they do not profit from it. Even more, their kids profit from this distance. Given that wealthy people are powerful in the US, this seems to be a more likely reason why public schools are not well funded.
So economically speaking, private schools on a large scale could be damaging to the quality of general education, much like a fragmented insurance market may eventually split into high and low classes.
Countries like Finland have public education on-par with the best private schools elsewhere. One component may be funding for education generally. But it will probably also be true that there is more funding towards public schools in relative terms, since powerful actors (in any sense) have an incentive to make schools for their children decent. In the US, wealthy people in general have zero direct incentive to fund public education.
Therefore, it seems to me that the solution is not to introduce competition. Speaking with US teachers and administrators, it doesn't seem that they could provide better education if incentivized by better private schools. What I hear instead is that public education is simply underfunded in the US. That would speak for the argument above.