I completely disagree, Yellowtip. Oxbridge is always inundated with applications from the creme de la creme from around the world. If your personal statement can show leadership skills, team work, sports and still achieve the top grades that will set you aside from the candidates who cannot show social skills, the ability to multi-task
You might disagree, but yellowtip is largely correct. In the UK, at the elite university level, we're chiefly interested in potential academic ability and excellence: usually demonstrated by achieved results. This is by no means a perfect system, but it's what we work within now. How a better system of selection to university might work is another thread. US teen movies & television series tell another story, but that;'s the US.
What you're doing is confusing results with causes. High achieving children, in families that value the development of human potential, are usually involved in all sorts of things, and generally excellent in a number of them. In a family where learning and doing beyond the usual round of eating, sleeping, working, and watching television are valued and encouraged as a part of life generally (ie not just limited to schooling) for all family members will often also be families where there is a high level of school-based achievement.
And so it might look like universities select with extra-curriculars in mind, but it's not usually the case (well not in the RG universities I've worked in all my life). It's that the "best and brightest" are also just generally more energetic, more interested in the world around them, more engaged in pursuing a range of activities generally.
So
if the uni is trying to decide on their last spot between two straight A candidates, surely the one with extra-curricular activities is going to get the place?
I can answer, No.
There are always differences between candidates, and ways of ranking them via interview, and other selection activities. Having the DoE badge, or being school captain, are nice, but IME in the UK they don't get you a university place. It is different in the US at the Ivies, but for example, in Australia, in a post-qualifications admissions system, it is the numerical score at matriculation (HSC or similar) that gets you in. Little else.