Before you read this, sit down and take a deep breath. I'm going to be controversial.
I say exclude all religion from schools. It's a private matter for the home and family. No hymns, prayers or special clothes and definitely no "faith" schools.
I agree with you on all counts. On the other hand, I absolutely loved hymns and psalms and things when I was at school, but it was the excuse to sing that I liked, not the religious thing. I'd have been just as happy if we'd sung non-religious stuff - happier, probably.
The ring thing I find ludicrous, sad.