The key thing is that in the US there are two distinct kinds of religious schools: indoctrination focused, like "evolution bad, women stay in kitchen" level in some cases, focused on "the Bible" and anti-secularism, and these usually have an academic cap because they're not going to do advanced math or stem, or teach accurate history, because of the focus on religion.

On the other hand you have more traditional religious private schools, whether Catholic, segregation academies, etc., and these kinds usually have good academic standards and are probably more comparable to what you refer to. They are also rarely intersecting with the charter system.