We went to look at Bancrofts. We live in the area and hear these rumours all the time. Our children go to a local state primary. Based on what I hear, what I've seen and the families with children actually there that I know, I would say that the pressure probably lies more with individual families than the school. I think parents attracted to the school in the first place are ambitious and focussed. Most of the time this is probably fine, but in a few cases the dear children can't meet parental expectations and it all falls apart. I think Bancrofts is a symptom, not a cause.
As for local prep schools, Loyola is the best academically, but it's boys only. Woodford prep and St Aubyns have good reputations. I don't know anyone with academic aspirations sending their dc to Oaklands, but I don't know many people sending their dc to Oaklands in the first place.
As for neighbourhoods in the area, Buckhurst Hill and the Monkhams and Knighton Estates in Woodford Green are considered more " genteel," than Loughton or Chigwell which get more bad press for being new money, etc. I think this distinction only means something to locals. Everyone else just sees all of West Essex as a great big hot TOWIE mess, lol.
As for race, deep intake of breath here, it is mostly white British. Although, if you dig a little deeper a LOT of people here have Greek grandparents or a Jewish granny, etc. maybe that's why people seem generally friendly to new comers. There are Black and Asian families here who seem to be completely part of the community. A few families each year from our state primary send their children to one of the three local private high schools, Forest, Chigwell or Bancrofts. They do so expecting their white children to be in a minority, albeit a large minority! I am unaware of any official stats, but this is the local presumption. TBH, I haven't given it too much thought because I'm not too concerned about the races my kids are mixing with, values matter more. I think if I choose a school, I will probably be self selecting into a group of parents who share a lot of similar concerns to me and values. However I am white, I can see how if you are from a minority group you would be concerned about making sure your child doesn't end up feeling like a token.