I don't want my children to be small-minded or fundamentalist, so I haven't given them a Jewish education. They have been brought up without any religion; they are free to choose whatever beliefs they like. I try to live by the words of John Lennon:
Imagine there's no countries / It isn't hard to do, Nothing to kill or die for / No religion too, Imagine all the people / living life in peace...
Is there anything more important than that?
I admire your passion and idealism. You have obviously given some thought to your children's moral future, which is a credit to you. But I don't see how the philosophy you have espoused is any less closed-minded than fundamentalist religion.
You don't want to force your ideals on your children. But by denying them their spiritual heritage, you are forcing your ideals on them. They are missing the chance to explore their Jewish identities during their formative years. They didn't choose that, you did. You have decided their religion for them. They are Lennonists whether they like it or not.
And if that song is your bible, then they are being brought up in a much more closed-minded religion than Judaism.
You have only quoted one verse. But I think the last verse of the song is the most revealing. It proclaims a worldview as parochial and insular as the narrowest sect:
You may say I'm a dreamer / But I'm not the only one.
I hope someday you'll join us / And the world will live as one.
So, according to Lennonism there is "you" and there is "us". You are the unenlightened ones. We have found the truth. But hopefully one day you will see the light and become one of us too. Only then can the world finally live as one. Is that open-minded?
Contrast this with Judaism's view that not everyone has to be Jewish. A non-Jew can live a perfectly fulfilling and meaningful life while remaining a non-Jew. They don't have to join us to be considered a good person. If anything can make us truly live as one, it is the recognition that we are all created by the same G-d, but we don't all have to serve Him in the same way.
We each choose a value system to live by and to teach our children. Whether you call it religion or something else makes little difference - it is a particular way of looking at the world. Can you imagine a religion that teaches its children to be proud of who they are, but that not everyone has to be like them?