Hi OP. I suspect I don’t live far away from you 
. I’m an ‘immigrant’ too, but have lived here 20 years. My kids are all teens and we have survived, so here’s my retrospective perspective!
What feeds the hyped up vibe in these parts is that you happen to be surrounded by the top independent schools in the country. Within a few square miles you have St Paul’s (Girls and Boys, Godolphin and Latymer, Latymer Upper, etc. Then there’s Hampton and LEH they can get the coaches out to. There’s Kings in Wimbledon and the GSDT schools such as Putney High or Wimbledon High, not forgetting Francis Holland Sloane Square and NLCS, CLGS, CLBS and Westminster if you’re prepared to travel a few miles north or east. Look at the Best Schools Guide and you will see that many (most)? of the schools listed in the top 20 are local to you.
Competition for these schools at 7+, 8+ or 11+ is ferocious and stressful for all concerned. Children are sitting at least 5 schools each (so five different sets of exams / interviews). Some are sitting up to ten schools. Nowhere else in the U.K. carries on like this - not even remotely. The reason some parents may want their children to have certain music grades or ‘extra curricular achievements’ is that it can help with school entry - ie they have more to talk about in the interviews.
Some schools have 15 applicants for every place. Even the so-called ‘second tier’ schools are more competitive than anything in other areas if the country because everything is so ramped up.
So it depends what you’re looking at. If you’re thinking independent school, it’s naive to think they can just waltz in, to be perfectly honest. If applying from a state primary, you will need to be proactive in terms of helping them with exam technique etc.
As for bikes and coats or whatever, I wouldn’t get involved in any of that. It’s irrelevant.
Re- music grades. Two of mine were quite musical but they didn’t do the all the grades systematically through school. I think they did Grade 1 - then they just did the lessons through school and then did Grade 8 just before they left. It does carry UCAS points. Some unis are more interested in this kind of thing than others. But the main point is, you don’t need to work systematically through all the grades. They do need Grade 5 music theory to proceed higher with singing / instrument grades - but rather than banging away at that when they’re young, mine did Grade 5 theory from scratch at school in less than a term in about Year 9 or 10.
There are some scary ‘sports mums’ (not to mention the dads), but your kids are either into this or they’re not, Just be led by them.
It does seem crazy - and that’s because it is crazy, frankly. But on the plus side, these kids won’t be phased by competition when they get to uni and they’re up against the entire world because to them it’s just normal. Being told you’re a genius in a low-performing school or in a less populated / competitive area can also mean that these children find they have an unrealistic sense of self and struggle when competition hits at uni or in later life.