"There are no secular/atheist private schools in Louisiana"
That's incredibly unlikely to be true. Have you actually ascertained that? If so -- WOW. I'm from Texas and they're all over, so I guess I would have assumed that was general. (Secular, I mean, not atheist specifically).
At bottom, the school voucher system is about giving parents choice. Just look at how secular many of the movement's leaders are if you don't believe that. Of course, sure, I'm sure various people have all kinds of reasons for supporting it.
Finally, as for "the principle is still applicable" -- how do you figure? The Court has never held that, and I am unaware of any good argument that it should. (Yes, separation is required in many places precisely because it ends up supporting a specific religion. Here, it manifestly does not. For example, there are Muslim schools in Louisiana, and if you wanted to start an explicitly atheist one, nobody would stop you).