I have friends with children at both.You are right that they don't give out 100% scholarships as you say, they share it around, but they do have a significant bursary system outlined on the website.
Realistically you are looking at 20% - 30 tops.The boys who got the the sports scholarships at Whitgift got 20% and played at national level, ie picked for England youth squads. The academic scholarship again just amounts to the same.
In music I know that the standard is very high.I know that the 20% again would be availiable to a very musical child.Usually we are talking about grade5 + preferably in more than one instrument.
However I am sure they could spot raw potential somehow too. I only know one child offered an academic scholarship and he was bright enough to pass for the grammars but by no means top of his primary class, so clever but not a genius.
Suppose I am saying don't be put off.You can only try.
You have nothing to lose by trying and do check out the bursary info, I think it went all the way up to 70 K per annum.