I am a lecturer (1994 group) and my current administrative role is departmental UCAS selector; that is, I go through the hundreds of applications to our department and select for interview or rejection.
The answer to the OP's question is, predictably, yes and no... 
Because we interview, the first round of the selection process is basically an automatic one, ie. I go through the pile, and those who meet our A-Level (or equivalent) entry tariff (or are predicted to) are invited to interview; those who do not are rejected (unless there are serious mitigating circumstances indicated in the candidate's reference (not PS - after all, they could lie) and even then, I would scrutinise their GCSE results to see if they have form). At this point, then, the PS is virtually irrelevant.
At interview stage, the PS can be important, though not always. Personally, I only use the PS to probe a candidate's ability. So, they say they've read X, I quiz them on X and related topics, etc. If I can, however, I generally ask them to perform a task on the day, and only turn to the PS in the final minutes of the interview. I never, EVER ask about their extra-curricular activities, unless they strike me as particularly intellectually interesting or relevant.
The thing is, in my opinion, the PS is only a useful document for analysing a candidate's writing ability. I have no way of proving that what they state there is true. They tell me they have Grade 8 in violin, or have done internships for KPMG, or whatever. How do I prove this? And why should I care?
I've said it before on MN, but I am NOT INTERESTED in a candidate's general all-roundedness. I want to know if they can study my subject. Whether they play international rugby, or have led their YE group, or whatever, is meaningless to whether they are up to the academic challenge of my subject. Really it is. I can see that in some more vocational subjects, some work experience, etc., would be useful and maybe desirable. But even then, the interviewer will still want to find out if the candidate can hack the course. Frankly, being captain of netball is no predictor of being good at philosophy, for instance...
However, I think that what I have just written primarily applies to places which interview. Probably those that do not, do look at other achievements as a way (totally random and utterly UTTERLY unfair - after all, poorer kids have fewer chances to do all the extras which parents typically pay for) of discriminating between otherwise academically identical candidates.
Personally, I think the PS system has become increasingly out of control and unfair. It discriminates against those with fewer opportunities, regardless of academic ability, and has started an extra-curricular arms race amongst parents who have - or feel they ought to have - the resources to flesh out their children's CVs.
I have interviewed for several Cambridge colleges (humanities subject), and again, the majority of my colleagues and I looked at intellectual qualities, and not all the added CV bits. We couldn't bear it when half the PS was wasted on twaddle about captaincy of this or that.
However - admittedly - older colleagues (ie. those who hadn't been to recent diversity interview training & had been at their colleges for decades ) often did, regrettably, find themselves impressed by that jolly chap who played in the rugby sevens, or that lovely young girl with the short skirt (I have seen this in action, I'm sad to say).
And the thing about the Oxbridge process is this: the subject interviewer will (consciously or unconsciously) be thinking, "Can I supervise this student? Do I want to spend several hours over the next few years alone with this student in discussion?". Things like personality, looks (not just looking smart, by the way, as so many on MN seem to think...), can sway it as this point. But again, it's not predictable in the way you might think. The physics geek who was picked on at school might have inbuilt bias against rugger-buggers. Who's to say that he hasn't been deselecting them at his college for years? There would be no real "trace" of bias, if ever other candidate was similarly qualified and there were only 6 places for 20 applicants, if you see what I'm saying...
Schools, and self-interested idiots like the chap who wrote this article, perpetuate so many myths about the selection process. Nothing we do seems to change this, sadly.
Sorry, long and rambling message that does nothing to help parents or students.