"Grammar schools should be for naturally academic children."
Mmmm, define 'naturally academic' or rather, if you could ever get a true picture of academic entrance to these places, try and isolate the number who get accepted WITHOUT extra tutoring or a private prep school education. I think you'd find very few.
I don't think an 'uncoordinated' child is a proper comparison with a child you would possibly politely define as 'non-academic' unless you are trying to suggest that such children are somehow equally incapable of ever being able to achieve - in this context academically.
Let's use your sports college analogy this way.
Your 6 or 7 year old is not very good at football and perhaps doesn't show any inclination to play it. However, your society prizes football above all else and all the local aspirational parents think so much better of you and your child if your child is good at football. You feel you are letting your child down and limiting his life chances if you don't ensure he plays football in the same way as everyone else.
The best players at football get in to the 'football academy'. Wow, what a dream come true!
So, you get your child playing football in his free time and hire a professional coach to do train him as you can't possibly let him 'fall behind'.
Lo and behold, at 11, he passes the 11+ football scholarship.
Some things are natural, some by achieved by graft. It depends if you think it's worth it and what you feel it represents. But don't let's pretend it makes you necessarily more of a 'natural' talent than anyone else.
Trouble is, out in the real world, not everyone plays the same game.