Hi - FYI, the GDST group of independent schools already have very competitively-priced fees. They also have various generous bursaries for able state-school children.
My daughter gained a full fees HSBC scholarship from Year 7 which also included £1000 for uniform & books over the 5 years she was at the school. She had to take an 11 plus type exam and we also had to complete a form each year to show our family income.
I know that GDST also do their own scholarships which will be topped up with bursaries, if necessary, for able children, including Art, Music etc.
The key here is "able". Obviously, it helps to boost the independent schools' league tables to have some very bright pupils taking GCSEs and A-levels & they are prepared to give a certain amount of money each year to those who wouldn't otherwise be able to afford a private education.
Other schools give a % of the fees - you just have to check.
You normally have to pay the entrance fee anyway but you would request the scholarship/bursary at the time of application. Then, if your child is accepted, they will be told if they qualify for a scholarship/bursary.
Hope this helps