I think the grief for parents would be much less if school work came into it and not just 2 hours of tests on one day,
Kent used to use just this system a few years ago here in Gravesend. Children left juniors at 11 then went to local secondary. The Grammar schools took children from 13 who had been assessed by their secondary school teachers as 'Grammar ability'.
I didn't live here then but my neighbour tells me people still talk about how their sons/daughters were cheated out of Grammar places by useless/unprofessional/spiteful teachers.
The system then changed to take the 11+ like other parts of Kent and it is considered much fairer because the child themselves is tested. Unfortunately tutoring does still lead to an unfair advantage for many but it seems the current 11+ is a case of 'better the devil you know'.
When I took the 11+ in 1978, (in a Bexley school but I lived 1/2 mile away in Kent) I didn't even know I'd taken it because it was kept so low-key and certainly couldn't be tutored for.
Also, the parents of 11+ passers like the system, but parents of non-passers say it is unfair.
I really don't know the answer!