Private schools, are under NO obligation, to take ill qualified applicants. The good ones, are far too expensive , even with NO taxation, for those in public school to attend. Private schools know exactly how many studenst will be attending their school, come the start of the school year, and have NO openings, for displaced public school kids. Your theorizing / speculations ; therefore, are not only from a totally ignorant position, but incorrect at EVERY step of the way.
The close of every public school, without warning, would not only engender utter chaos, as far as the children are concerned, it would also cause a serious ecconomic problem. It takes longer, to find a building, hire teachers, buy supplies ( you STILL haven't said where THAT money would come from ! ) , and sign up students, than you suspect. A free market is NO answer to the question. You aren't familar with schools, teaching methods, nor the opperation of schools. Even IF It would only take a few months ( doubtful ) , the displaced public school children would be in limbo. Don't, for a moment, believe that all of the parents would home school those children. Unfortunately, quite a number of the effected parents, n matter WHAT you say about those now being home schooled, would be incapable of dealing with doing that. Remember, there are NOT motivated to do it now; they wouldn't be motivated to do so in this hypothetical case.
The fired teachers, would be EXACTLY those, who would be again teaching in the new schools. I doubt that YOU will quit your present job, and teach them. So, even WITH a free market ( and I am a free marekteer ) , this is a zero sum game, and a straw man argument from you.
What would I do to fix public education ? Go back to the way it was , from 1900 through 1940 . Throw out All of the programs, ow extant in education departments, and reintroduce Normal School methodology. Burn ALL of the current math, English, and history texts, and use the ones, from the above era. Remove EVERY " diversity progran, ALL " touchy / feely " crap , and reintroduce geography, and history ( as TWO distinct classes ) , bring back the teaching of economics, at the high school level, AND the GREAT BOOKS series. No more " HEATHER HAS TWO MOMMIES", at all.
Use ONLY English emmertion classes, and close EVERY ESL program down completely. Uncouple the mainstreaming programs, and get the emotionally disturbed, extremely disabled, and the menally retarded OUT of every school, and every classroom. Put those children back into highly specialized schools of their own.
Reintroduce being left back, skipping, doing three years of middle school, in two, and put back into place the OLD birthday cut offs. Little boys, are now being held back, for NO reason , except that the kindergarten and first grade teachers, are ill prepared to handle them, now.
I'd also get rid of the UNIONS; however , that can't be done in any legal maner.
IN the " bad old days ", and yes, they were NOT as " good " as those who know only fantasy history think they were , there were many problem schools, and problem children. Most of them were in rural areas, NOT in big cities; surprisingly enough.
Now, about FR. In certain circumstances, yes, this IS very much an " old boys / girls " place. The reason for that is, most newbies no longer take the time to read the archieves anymore, just read the threads, and get to know who is who, and what is what BEFORE they throw themselves into posting. Newbies now accuse other posters of believing something that they don't / being what they aren't, without any justification. They ( YOU ) are also far ruder , when replying, than most " oldtimers " are; though, when trashed, they ( I ) will give back in kind.