Are grammar schools better for above average children?

From personal experience, yes. Going from primary to secondary grammar was life changing. From being ignored because I could do the set work easily to actually being expected to think. My self esteem was rock bottom at primary, the work was fine but teachers weren't interested or actually were pleased to show other kids my work when I got something wrong. It improved dramatically when at a school where I was stretched and rewarded for effort. However, that's just my personal experice.

The tops sets of a Comp and the top sets of most grammars are interchangeable
they are all expected to get A*/A

and good comps push their bright kids really hard
as do grammars
and private schools

DS1 (and soon DS2) goes to a single sex selective secondary, but went to a mixed ability primary, so as a parent I've experienced both.

These were our reasons for choosing a selective secondary over the mixed ability alternatives:

I like him learning with students who are in the same ability range as him, because they drive each other on to achieve even more. In mixed ability primary he was always top of the class so got a bit lazy.

I like the single sex aspect because it is cool for boys to work hard and achieve. In mixed ability school he was the only boy who was very able, and got teased for it. Being high achieving was seen as a thing for girls.

I like the fact that the subjects on offer are suited to his aptitude: Latin, economics, further maths etc. He has no interest in technical or vocational subjects, and the mixed ability school is more skewed in that direction.

I like the fact that every student wants to be there and wants to learn. He says there is very little disruption in the classroom and high levels of respect for the teachers. In the mixed ability school that many of his friends attend, a small minority of children don't want to be there and sometimes disrupt learning for the rest.

Selective schools aren't the right environment for everyone, but they do seem to work for some children.

In the bit of Kent I'm in they mainly are super selective.

If my children were in the state system I would rather they were at the grammar schools than the comprehensives. They generally produce better exam results and the children (or the parents of the children) are more aspirational. There are exceptions, obviously. The facilities are slightly better, although i'm unsure whether class sizes are similar.