The difference between Tiffin selection and that of independent schools is that the latter have more freedom to look for "potential". So they interview, can look at approach as well as raw scores, can look at what school a child is coming from, references, other contribution a child may make. (My guess is that achievement in music or sport suggests a child who can also get their head down and achieve academically.)
There is a lot of difference between heavy tutoring and preparedness. Parents whose children are coming from the state sector do need to look at what independent school exams expect and make sure their child has covered the ground and is used to doing tests under timed conditions. However kids are practicing Grammar tests for years and years. I remember one mum whose very bright daughter was a Bute, being told that there was little point in her taking the Tiffin exams. None of their pupils got in. Yet this was a school which regularly gets a huge number of SPGS offers.
Yes tutoring happens in Private schools. Before AS, DS and his friends were making plans to have a day off revision only to discover one of their friends had three seperate tutors coming in that day. The others were astounded. (And a cautionary tale, this was the boy who in their friendship group did least well at A level, in part perhaps because he seemed to lack confidence in his own abilities and became very nervous.) He also discovered that an overseas boarder friend, who he had considered exceptionally bright, spent every Christmas and Easter vacation staying at a tutorial college.
That said I suspect the proportion of tutored kids at private schools is lower than at Tiffin. Ron is right that high aspirations and expections of peers and peer's parents will have an impact, though my view is that this can be positive as well as negative. DC will come away believing anything is possible as long as you work hard and get the results. The problem we have spotted is that it has taken a while for DC to realise they are actually better than they think at maths and science. So DD was not top set until Yr11, yet is now being told it is a real shame she is not taking double maths. Ditto DS who is finding he has a higher place in maths exams at University that he ever had at school, and again probably could have considered specialising in maths or STEM. We had a bit of this at Prep, where the projects on display were usually those we knew had been produced by tutors, nannies or parents. In some ways this is fine. Better to peak at University than before. However at any academic London school it is wise to keep reassuring DC that they are doing well, even if day to day they feel they are being outshone.