I think intelligence is far more complex than most people think. With the right experiences it is possible increase a child's cognitive ablities. The brain becomes more efficent with use.
A big issue is that private school children take a disportionate number of places. Partly because the richer experience of a private school makes children more intelligent and partly because of better prepation. It would help if private school children had to compete for say 7% of places at a grammar school. That way state school children would not have to compete against private school children.
Maybe grammar schools should have some financial banding. Ie. reserve say 10% of places for children in band C housing or below. That way children from poorer backgrounds whose parents cannot afford tutoring would have some chance. I'm being arbitary as it varies across the country what a band C property would give you.
I agree that there should be transfer of children who are blatently not grammar school material to the secondary modern and vice a versa.
Frankly the eleven plus system is fundermentally shit. Children vary so much in their development. I would rather have the divide made at 13. Less academic children may well choose the BTECs without someone saying they are failures.