I think it's funny that so many comments about birth being 'private' don't acknowledge the fact that most people experience it surrounded by strangers.
Personally I'd rather have my child in the room while I gave birth than some 19 year old student doctor. My child watches me have a bath, sees me naked and wanders in when I'm on the loo. So far I've not noticed him being traumatised by the experience.
As for children becoming distressed - well, that's why you have someone there: to whisk them away if they get upset. But they don't usually from what I can tell.