99% of the private schools in my state have a religious affiliation, which is one of the reasons we opted not to send DS (or DD for that matter). They do attend church regularly (with my parents), but I felt that *I* would have issues - yes, sad, but very true.
Anyway, there are students who do attend these private schools that have different or non-religious backgrounds, though I live in the Bible Belt, so they are rare. But I don't think that there is anything wrong with putting your child in one.
Also, I don't know where you live, but some states do not have early entrance to K, but have a grade acceleration process once they start K. DS6 was lucky enough to be a part of this "highly secretive" process.