The short answer is "no", there are not proven alternative models, even without further specification.
When one specifies that one is limiting the alternatives to "private, for profit, schools", this is even more true. The vast majority of private schools (all religious schools and almost all secular private schools; keep in mind that charter schools are government schools although sometimes managed by an outside company pursuant to a contract with the public school system) at all levels are "non-profit".
"For profit" educational institutions, pretty much across the board, grossly underperform relative to non-profit and government run schools, and there a very few examples indeed of high performing for profit schools. For example, none of the schools identified by @user4012 are "for profit" even though many are private. In higher education, for profit schools are the bottom feeders of the marketplace and charge high tuitions for exceeding poor results.
In K-12 you see basically the same thing (i.e. mediocre performance by "for profit" schools), except that "for profit" K-12 schools are much less common and have a much smaller market share than "for profit" higher educational institutions.
There are private and charter schools that consistently perform well relative to public schools with comparable student bodies, as other answers have noted. Almost every major metropolitan area has at least one or two of them.
There aren't very many of them (certainly much less than 5% by either number of schools or student body proportion in a metro area and realistically closer to 1%-2% measured by share of overall students in the age group in the area).
There are very few cases indeed of the good work of those high performing schools being reproduced in a consistent and reliable manner. Often a flagship school will perform well but efforts to make another school on the same model do not achieve the same results. Even if it can manage one or two successful sister schools, finding the "secret sauce" to consistently mass produce these high performance schools has been pretty much completely elusive.
Often, sustained high performance involves factors not available to competing public schools or to the educational system as a whole. Some of the more common factors, include, but aren't limited to:
a. Requiring more time from students than other schools (longer school years, longer days).
b. Engaging in behavior modification programs that can't be implemented without full parental and student consent and buy-in, which mandatory public schools can't achieve.
c. The right to swiftly remove students who are not succeeding in the program. This can work in a particular school but not for the system as a whole as there is an obligation to educate everyone.
d. Financial subsidies from some source not available to other schools (e.g. religious subsidies from a sponsoring institution, endowments, high tuition, corporate loss leader investments). It's amazing what one can do with money although it translates much less directly into outcomes than most people would intuitively think.
e. Securing outstanding educators and educational administrators by whatever means possible, that are scarce in the labor markets or educators and educational administrators (i.e. maybe there are 5 people in 100 who make their living as teachers and principals who are profoundly more competent than everyone else in the field and whichever school gets them will perform well, but there aren't enough of them to go around).