I think you missed my point.
Certainly I'd rather put a child in a private school that any public (government) school. But is your wife's school designed to prepare children for public school?
I didn't miss your point, that's why I put "public or otherwise" in parenthesis.
The academic demands of public kindergarten are typically much lower than those of private schools, so a child who attended a good private pre-school should be better prepared for either a public or private kindergarten.
Is there some specific difference that you are alluding to that requires a different type of preparation for public K?