This one is the Frankenstein type idea. The last one was my own.
It is likely that with a voucher type system (I'd prefer a tax credit system rather than a voucher, the tax credit is more individually flexible and evades the goofy problems of private schools often being parochial, for now), that competitive schools would emerge to challenge the public school monopoly. Right now with tax dollars forced to go to one location, there are no feasible economic reasons to create additional/different schools to meet a prospective demand for such. Only those parents capable of meeting through their own independent means fund their demand for a higher standard of education, hence only two private schools (the back and forth responses indicated there are only two private schools in the other fella's home state). If indeed we are speaking of collections of people so small that a demand would not be economically viable, then there is a problem. There are few such communities left. The answer is not to get better teachers only in public schools, but to have better teachers available period and have parents/students able to select them through a market process rather than through a (union controlled) license system.
This was unsurprisingly met with the resistance that is deserved for supporting vouchers even partially. "When vouchers have been implemented, it has been shown it works as a tax cut for the people already going to private school. The problem is they can accept any one they want. They can not let in poor people or minorities or any kids that achieve at a low level. This would cause the public schools to be the schools of last resort and only house the lowest achieving children, and no teachers would want to go there. This downward spiral, i fear, would create a illiterate class in our society, something much worse than the status quo." So I had to defend my actual position: tax credits used to create a totally private market for education. My position is unfortunately too radical to be popular (much like privatizing OASI), but I have no problems with that.
There are several fundamental problems with this, but since many of them are caused by my previous vague preferences, I must be more clear: I very much prefer a tax credit system to a voucher program. A voucher leaves in place the current problems and places at their side an option which doesn't really offer much. A tax credit does more what I want; kill off the public school system monopoly and create a marketplace for education.
For my purposes the essential problems are that there are 'government/public schools' and 'private schools'. My interest is that there should be 'a school' or 'a better school', or 'an affordable school'. There should be no "public school system" at all. This is not to say that there would not be (extensive) public subsidy involved, but my preference would be that the subsidy of education is actually executed by the free choice of individuals given tax credit by government(s) for spending money to foster educational growth in others. No preference should be exercised for a state monopoly on education as this severely limits the ability of individuals to supply education for their progeny.
1) The current status quo you are defending is precisely the spiraling scenario you are afraid of. Some public school districts are ALREADY a school of last resort whose students are generally poorly educated (if they are still attending) and whose districts are generally composed of the poor or minorities. In strict terms, families are stuck there because they cannot execute the choices available to them to improve the education of their children. They cannot afford a private school, even with a voucher system, and they cannot afford to move to a more affluent neighborhood with a genuinely useful public school system because the property values are much too high. Their children must suffer with what they have, and in many cases, suffer they do. Under a non-monopoly system, there would be options available to them to spend whatever funds they can desire to do so for education and a basic requirement that those funds afford for their children at least a minimally competent education (most essentially issues like basic math and literacy along with a smattering of civic awareness, but certainly there are many other issues which might be considered of value to the public utility of education). There is no accountability in the present system owing to the government monopoly over school systems in many locations and we should expect no responsiveness, adaptivity, or improvements so long as a monopoly remains firmly in education.
2) As previously discussed, a person without cannot make these choices. But to put it quite plainly, a person with means can already perfectly execute the system you are describing as happening only under a voucher or tax credit system. They can, if they choose, send their children to 'a better school' by moving to an affluent district or paying for private schooling. If it is their wish, and unfortunately this is still true of many, they can send their children to a school with very little in the way of minority attendance (and that very often more or less only Asians, depending on local demographics). A monopoly is naturally more discriminatory than a free market as it imposes fewer internal costs on discrimination (of any prejudicial format), where as the market imposes the costs of discrimination directly. A system which allows favorable preferences to be expressed for higher values placed on education ONLY by people of means is necessarily unfair for everyone else, as is a system which naturally excludes minorities. Such systems impose a social cost of vigilance against that minority or illiterate underclass and deprivation of the diverse contributions and scholarship potentially provided by such. Education in its pure ability can become a meritocracy where people can achieve through their innate ability or interest, regardless of means or race. Right now it does not achieve this end. I propose that it would most nearly do so under a responsive market circumstance and propose to abolish the state monopoly, not merely place alongside it an attractive option for people already able to manipulate it to their advantage.
3) I am not sure that I understand the significance of objection to accepting or rejecting 'low achievement'. The issue of education should be one of supply and demand. The demand of parents is for their children to get an education, generally speaking, and a supply will be made to meet it. Even low achievement students will therefore have options, such as schools or teachers which specialize in such 'problem students' or special needs (a name which should really apply both to students of minimal or exceptional ability). As undesirable as these options are in the present circumstances, they can be accounted for. Teachers might expect (and receive) better pay and have better administrative support in exchange for tackling these problems. I would also presume that the necessary involvement of parents could improve where they are more responsible for the decisions of where their children will attend and where they are aware of the immediate costs of that attendance (even if they are re-compensated). This factor can be overvalued but is of some consequence. Right now it is of no significance.
Or alternatively/alongside we could have more practical educational options combined with a re-accessible education system all along its length, at least beyond primary education, which should probably be mandated to about 7th grade. Either high school or the first two years of college are a joke, and many people shouldn't be doing both. Nor am I at all ready to be convinced that a high school diploma is somehow essential. In any event, academic achievement is often, by itself, useless except to other academics. The assumption that all students should be at least adequate academics is a value judgment which isn't realistic. Not everyone likes reading Shakespeare or studying WWII while commenting on politics, as I do. If we assume that they should be competent in such things in order to be considered literate or educated, then I have to suggest a functional disagreement with what the aims of education should be. Namely that it should provide a desire and ability to learn, provide some basic tools to do so, then to help feed whatever hunger for knowledge and wisdom that creates. Naturally some subjects will be disagreeable to the individual or their ability and they may be of low achievement in these arenas. I'm not how sure that's the state's problem to resolve to the exclusion of other agencies and, more importantly, without the option of choice by the individual and their parents on how to resolve it.