The only significant non-religious belief systems I can think of which fit the bill are nationalisms and Marxist-Leninist communism

I'd add mock-heroic free-market capitalism to that. I suspect a lot of the hatred of 'socialism' you see in the US is exactly from an equivalent level of indoctrination.

In fact free-market capitalism is the state ideology of the west, and increasingly also other parts of the world. Instead of a catechism, you have advertising and state propaganda. Instead of priests, you have talking heads and dumb headlines.

Even so - it's all-pervasive, and it's almost impossible to shield a child from it. (I suspect in the UK it actually is impossible, and parents who try will have their children removed by our Social Services.)

The older religions are largely misdirections which keep people distracted an unaware of the extent of official indoctrination.

A further issue you don't seem to be taking into account is that a lot of personal views expressed here aren't inspired by the viewpoint of an atheist teacher but the viewpoint of a one-time child who suffered through the 'teaching' of theist parents or school teachers.

"Inspired" is beautifully put. How am I to deal with views expressed on religion that in reality are views on abuse? Is it really asked too much of the authors to make clear what statement is inspired by what? I reject the notion that religion, not persons enforcing dogma or exploiting positions of trust, is abusive. And I reject the notion that embracing religion is the same (or related to) child abuse, forcing children, and what other insiuations have been made.

DoDo:

Now, what is your opinion about the political parallel? Would you approve of Free Democrat schools, Sunday instruction of six-year-olds in the basic tenets of Social Democracy at the local party headquarters, and the same children wearing party insignia to school? Should voting age be eliminated? Why should parties be more constrained than religion?

I think beliefs can be sorted by how close they are to the core of one's personality. Some political beliefs are overarching mere party politics, for instance the importance human rights have (or property rights). We all probably radiate our deeply held political beliefs anyway. I would recommend a bit of restraint in the case of party politics, but I don't think it is vital or should be enforced. Can you really object if some members of a party meet in order to organise something and take their little children? Probably not, but where is (realistically!) the difference between that and "Sunday instruction of six-year-olds in the basic tenets of Social Democracy at the local party headquarters"? By the way, I used to wear a SPD jacket (gift of my mother) to school, although there was (and is) a ban on party insignia in school. They couldn't very well undress me, so they rang up my parents who promised to remind me not to wear the thing to school. Which they did, and I ignored. The more hysterical the reactions became, the more I enjoyed it. And the sky did not fall down. I would even have joined the SPD if they had allowed under 16 year olds. By the time I was sixteen I had acquired enough political wisdom and no longer wanted to, so perhaps having such a minimum age makes sense (but not much).

When Merkel wanted to get out of the exit from nuclear power (before Fukushima), I took part in many protests. My daughter, then 10, asked me to explain what I did and why and then declared she wanted to accompany me. What would you have told her?