Mary, Mary, Mary. Do private schools get to choose their students? Yes. Do richer kids go there? Yes. Do they offer special ed or special services for children of immigrants? No. 
You say there are no teacher evaluations and a few words later say they are all evaluated well. Which is it?
I'd love to see your "research" cites. Any will do.
Fact is, our economy has led the world for 60 years with this horrible (your thoughts) public system. The overwhelming facts point to a very good system, but with very bad results in one area: educating the poor. This isn't a simple fix. Families, neighborhoods and other issues need to be addressed. Scandinavian nations and Finland faced the same problem decades ago and resolved to lift their poorest citizens up. Yes, these are the same nations that score exceptionally high, now. They also have far less crime than we. 
The bucks you save hopefully will make you feel very happy. You see society as a one-boat for one system. I'm sorry for that.