The whole Charities Commission brouhaha was the last government's ideology that was impossible to enact. It is extremely difficult for a school to either become a charity or to stop being a charity.
If you speak to anyone involved in finances in an independent school, they will tell you that overall discounts should not exceed 10% of fee income in a healthy school. There are plenty of unhealthy schools, financially, btw - but you don't hear about them until they start going tits up. A bursary fund of 5% is enough to satisfy the charity commissioners. This is unpublished information because it is only when a school presents its criteria that the commissioners say yes or no. Now we are a few years down the line from the original initial discussions, 5%, in practice, has been acceptable.
That is not to say that 5% is the level of discount to an individual child. That would be pointless, as the parents would still have to be very affluent to pay the other 95% Most healthy schools will give one bursary per year that is close to full fees (eg 75 - 100%). Some schools have additional income for bursaries, separate to fees, so they may be able to give out more.
Scholarships of 10% are marketing tools for the school, and not to be confused with bursaries. They bump up the fees for everyone to pay for these. If you look at the wide range of school fees charged, you can see which schools give out scholarships as if they were candy. Again, a healthy school will keep a tight reign over scholarships, and probably prefer to give exhibitions instead.