I think the boys and the girls schools offer a superb all round education and if the Beckham children can pass the exams and interviews they should go for it. They will get in and thrive if they are academic enough and also have a talent for something other than academe. Peter Hamilton is a very grounded headmaster. The girls school is very used to famous parents and is very discreet and safe. Haileybury doesn't cut it if a child is bright enough to get into Habs-I know this from a good friend who had sons at each.
I dont understand the Jewish comment at all- children from all religions go to Habs and they have different assemblies for each and my non Jewish children (and their friends) used to go to lots of different assemblies every week, depending on what the topic was, the food etc being offered and whether a friend (Jewish or not) was taking the assembly.
Class isn't a big issue at the two schools. No aristocrats go there. No royals. Lots of very wealthy. Also quite a few who, like both mine did, have generous bursaries. Terry Leahy's boy went there. Neither the girls or the boys school tolerate poor behaviour or sloppy work, regardless of background. One very wealthy bully got expelled.