Bubbles you are right that some kind of extracurricular engagement is a good thing. But you make two mistaken assumptions. The first is that it needs to be in the form of clubs. Someone who has demonstrably spent their spare time diving into the Queen Victoria shelf of the local library, or writing historical fan fiction, say, is going to be more interesting to a history admissions tutor than someone who plays five county sports (which to be honest would be a red flag for someone who wants to come mainly for the sports opportunities rather than the academics).
Secondly, the extra-curricular stuff is highly unlikely to be a tie-breaker. Academics want to admit bright, engaged, committed students who are going to be interesting to teach - who can engage, don't collapse when challenged and can think well on their feet. When deciding between candidates, it is those things that are the deciding factor. In a tie-break situation, tutors are much more likely to sit discussing the relative merits of two candidates' answer to a question about whether historians should be more interested in kings or paupers, or some such, than whether grade 4 bassoon is more entrance-worthy than 3 terms of ballroom dancing. (and also, if a candidate meets the level for entry, don't forget they will likely be pooled for interview at another college).
That's not to say that there aren't unfairnesses in the system - clearly there are. Public school candidates will on average be much more used to a cut-and-thrust intellectual questioning style of teaching, and therefore more likely to perform well under interview pressure. Narrow teaching to the exam is probably more common in state schools, which can also be a disadvantage. And like I said earlier, metrics (i.e. marking each stage of the process including admissions test, submitted essays, interviews etc) removes some of the discretion of tutors to admit promising candidates from disadvantaged backgrounds who might struggle with some aspects of the process. It's clear, for instance, that public school candidates are likely to have had more help from teachers or parents with their submitted essay than someone from an over-stretched inner city academy. It's very hard to see admissions can allow for that kind of thing. But Jocasta having an advantage because she can say in her PS that she reached county level in dressage? That really shouldn't be the focus of your worries.