Wow - I think you've got your history very mixed up there. Those little red schoolhouses were few and far between. You didn't choose which one to send your kids to based on their performance - you hoped there was one close enough to send your kids to. And there would be ONE that was close enough.
Mostly, those one room schoolhouses were successful because they weren't trying to do nearly as much as schools are asked to do today. There were no extra-curricular activities, no football teams, cheerleading squads, chess clubs. There were generally no art classes, and usually no music classes - if it was a city school, and the parents were fairly well off, there might be a piano. They did not teach calculus, or chemistry.
And, depending on exactly what period we're talking about, there might absolutely be a school district consolidating some operations for multiple schools in an area.
But the real reason those schools were successful is that they didn't have to educate everyone. They could expel troublemakers, and those who weren't interested in being educated could generally leave school when they had had enough (even after compulsory attendance was introduced, it was often possible to get exemptions, at least if you lived on a farm).
Nothing against charter schools, but they are a different concept, with a different agenda, goals, and means of acheiving those goals, than the old one room schoolhouse.