Instead you have to address what happens to the 50% of people who don't get into the selective schools. You can't improve overall quality through rejecting students, because there's no "away" to reject them to!
People who are actually involved in education are more likely to tell you that the key factor for most of the lower-achieving students is obstacles, especially out of school; are they getting enough sleep and 3 square meals a day? Do they have at least one parent around who's engaged in their education? What obstacles does that parent face? Is the child being bullied at school or home? etc.
I live in a very rural area where a grammar school has been maintained (which my wife attended) and most of the people who go there end up at good universities and then move away for good careers in distant cities. Meanwhile, the people at the comprehensives usually end up staying in our area on lower paid jobs.
Does this lead to be a better society? It certainly provides a way for poorer but smarter people to find a way 'out' of a humdrum rural location, but I don't think it makes our area any better in and of itself.
Does this mean that such schools apply more focus to issues such as the ones you list in order to better compete?
I have friends who moved down to Kent so that they could get their kids into selective grammar schools. There's little evidence that these schools provide better educational outcomes in terms of progress. They skim the most able from the system at the beginning of the secondary school process and at the end, they are still the most able, though not measurably better educated than the most able who go to non-selective comprehensive schools.
Mind you my son transferred between private schools at age 16 and the interview was mainly about rugby...
[NB Both my wife and I went to very humble state schools but we could afford private schools and, to be honest, the quality of teaching and facilities at private schools is vastly better to state schools so I've never regretted it].
My own personal experience is of State, Grammar (selective) and Private. It's all about the intake, I feel. The quality of teaching isn't really different. If anything the Private school teachers are lazier because they don't have to try as hard. They also seemed more arrogant to me.
You might note that at what are considered "good schools", not only are neither of those things true, but trying to implement either would cause the school to be quickly abandoned.
However also anecdotally a friend taught at a private school for a couple of years. Parents made it quite clear that it was his responsibility to ensure their angels passed the exams, no matter how thick they were. He didn't stay for long due to that attitude.
Moreover, as money spend on high education in many cases is essentially unproductive tax, sending more people to universities may even harm the growth.
All I can offer is my story on this. I've lived in the UK all my life in all sorts of regions from those of lower working class (my hometown) to my recent years where I've lived in rather upper-class "lawyer-doctor-business owner" regions. The difference in education between the lower and upper classes is stark. Let me explain.
Upper-class children are enrolled in schools of like-minded high-achievers full of entrance exams and interviews (provides a non-zero floor to student "intelligence/quality" at the school), and strict rules/discipline. They are further pushed towards degrees leading to higher paying careers - Finance, Medicine, Law, by their families and the schools themselves. Their families purchase extra tutoring, they have smarter parents (this is a positive feedback loop across generations!) to guide them, and so on...
Compare this to lower-class children, who, in the UK (and i'm pretty sure everywhere else essentially) are enrolled in "state schools" (free). These schools have no entrance requirements (there is zero floor for student quality), the families almost never purchase extra tutoring, and are much less pushed towards "meaningful" degrees. The schools offer equal environments to the best and the worst of students.
This imbalance in education (and by extension upbringing in general) is what seeds everything else. And indeed, in the past few years the UK government has recognized all the above. How they plan of solving it? The reintroduction of "Grammer Schools"! These are state schools (still), but _do_ have enterance requirements, thereby allowing lower-class students of high intelligence/quality a chance to separate from the worst students in state schools and enroll in a more challenging education. There is much debate in the UK whether this will work, or will simply just allow children of rich families to more-often/further separate themselves from poor families. Time will tell.
That's my 0.02c anyway: Education.