I honestly doubt that the US system is any better than ours.
Although it may seem like a more well-rounded to be basing entrance to university on extra-curricular activities, but doing so has a much more complex impact on equity than what many people here are suggesting. For one thing, it's fair enough IMO for a university to be looking at your transcript and then assessing your intelligence. It's not without problems and we all know that top performers at high school may not necessarily be top performers at university, but it makes some degree of sense. But unless your extra-curricular activities are closely related to your career goals or your degree, how is it a relevant consideration when assessing your propensity to graduate with your degree, get decent grades and go into a career? (These are the things universities are really assessing applicants on)
Moreover, how can your entire life and all the things you've ever done be adequately assessed and ranked by admissions officers? Even if you could unproblematically say 'your set of experiences is superior to someone else's set of experiences', I believe that assessing people this way puts a bias towards people who have had the opportunity and the money to participate in extra-curricular activities in the first place - not every family can afford expensive fees for rowing lessons or musical instruments and such. Those applicants would potentially be inadvertently discriminated against on the basis that they couldn't do that wide range of activities.
Don't get me wrong, admitting people on the basis of academics alone isn't ideal either, but I think it's better than the alternative of making students not just freak out about exams but also about the worthiness of their entire lives.