I think you are still seeing this from a much more modernistic, compartmentalized perspective where religious faith is separate from everything else, like it's a dollop of whip cream and a cherry garnishing a scoop of ice cream. You can have that ice cream without the cherry on top, or with it if it's your thing.
But the reality of it is for most cultures which are traditionalist, and not modernist in nature, is that the cherry is fused with the ice cream. It's not ice cream with a cherry on top. It's cherry ice cream. It's not separated. The ice cream is the culture, and in the traditionalist world, that culture is a religious culture, or cherry ice cream.
So any of the normal cultural indoctrination that all human beings who are part of a social world (as opposed to a wild child), will either include religion or exclude religion, or religion as an option. But it's all indoctrination, regardless if religion is included or not. In a traditionalist culture, children will be programmed with a religious perspective of reality as part of the culture they are raised in.
Not teaching your child about religion, is only an option for a modernist or beyond which are able to see religion as a garnish, rather than a flavor of ice cream itself. But all of it, is indoctrination or cultural programming, whether it includes religion or not.
What I don't agree with here is to say that religious teaching of children is simply indoctrination. It is no more nor less indoctrination than programming children with any cultural values and worldview. All that is indoctrination as well. We can't single out religion as something different that way, especially when someone is part of a traditionalist society.
What the issue is really simply traditionalist views of religion as part of culture, versus modernist views of religion as a personal preference and option. The issue in our politics is really between these two camps, those that see religion as a cherry garnish, and those who see culture as cherry ice cream.
This is good opinion article I read this morning where you can see what I said in the things the person writing it was talking about in the split into our culture wars. What he doesn't touch on is how this is traditionalism versus modernity, in so many words:
Opinion | ‘Christianity’s Got a Branding Problem’