"Children's Centres were set up for every child aged 0-5 irrespective of background"
Yes they were supposed to be for everyone. The idea was that universal entitlement is less stigmatising and more acceptable to people who really need it.
They originally had a really broad remit to provide support and childcare generally and targeted support where needed.
Now there's always a balancing act to be struck with this sort of thing. Because while few working class people want to attend a stigmatised service for "people who need help"- neither will they show up to something dominated slightly intimidating middle class Mums.
I understand that some centres walked this line better than others.
IMO the project was compromised by budget cuts almost from the start.
Lots of centres had to cut the more general "everyone welcome" type projects and "extras" in order to concentrate on the most urgent child protection type things.
This had the effect of making the Sure Start Centres (already viewed as a bit "proper" and "worthy"- I remember lots of Mumsnet threads about them trying to feed toddlers raw broccoli and banning Mums from holding tea cups) look like an arm of social services to working class parents. Who duly avoided somewhere where "they watch you".
Meanwhile middle class Mums continued to attend on the assumption that child protection somehow doesn't apply to them.
Its like an object lesson in what happens when you try to provide a universal service on a targeted budget.