My dd and SiL are not at all religious, but will be sending their dcs - if they can - to a C of E primary which is not only the closest but also a very good and happy school.
The majority of children will 'unbelieve' soon enough anyway once they get older, unless the parents are religious, and often enough even when they are. Dh and I both gre up in an era when there was quite a lot of C of E at school, hymns and prayers every morning, plus Sunday School when little, and his parents did go to church, but it didn't turn us into remotely religious people once we were old enough to decide for ourselves.
But quite apart from all that, I don't think a bit of mild religion is a bad thing - so much of our own and European history is bound up with religion - it's all part of general knowledge, which they can be aware of without believing in it once they're older. (Not that general knowledge is very 'general' nowadays, if it ever was.)
FWIW my dds both went to an independent Catholic primary for a while - one for just one year, and one for 4. It was not from choice, but because we had just returned from a long period abroad and it was the only school that had places for them.
Despite a great deal of God there (more than one teacher was a nun) and having to write AMDG (to the greater glory of God) on every page of their exercise books, neither of them was 'turned religious' by it. It was incidentally a very good and happy school, and a number of children of non Christian faiths, including Muslims, went there. Since it was an independent, they were not allowed to be excused from any of the lessons on religion, but I never heard anyone complain. The vast maj. of parents were more concerned with the general quality of teaching. which was excellent.