A lot of
here.
With two children having almost gone through these schools I have known a fair number of pupils on bursaries at all these schools, full bursaries too. However the bursary pot which rarely comes out of fees isn't limitless and the way that it is divided is according to the most deserving as defined according to their ethos, often a founding ethos, usually those who would most benefit from the opportunity. They also use some of these funds so that they can continue to provide stability to pupils whose parents are no longer able to pay. Quite a few go proactively into state primary schools to encourage applications from pupils from the most disadvantaged backgrounds who might not even think to apply.
They are all actively trying to raise money to increase the pot but in the end it has to be rationed somehow. My DDs have friends who have been refugees eg from Sri Lanka and whose parents make ends meet driving taxis, whose parents have lost their livelihood as a result of Cancer or bereavement, whose fathers refused to pay for their education following divorce etc. etc. And even in the latter cases that emergency support ceases at sixth form. When my DD was sitting one of the exams a single parent father whose wife had recently passed away after he gave up work to care for her, asked me where he could go with his toddler and we walked to the park. His DS's teacher was paying their admission fees and fares to give the clever DS this chance. He was from Chessington where I gather the local schools are not great so I hope he had that chance.
Frankly after all these years of paying fees I wouldn't have minded at all if they had gone to be used in the way I have witnessed, and the possibility the money is doled out in the smallest possible chunks to get the cleverest is wide of the mark, scholarships maybe but that is not what bursaries are for.
I assume they would not write to say that you were not clever enough for a bursary to give you a chance to find the money because you have put yourself into the pool for them to consider whether your DC is the most deserving according to a number of factors, financial, in terms of family background and academically, of the opportunity out of those unable to pay and they are considering carefully all those factors. Frankly if you can find the money then you shouldn't have put yourself in that pot in the first place. It is a bit naive to think it is limitless, and will give that opportunity to anyone who gets through the admissions process on a level playing field with all those who can pay. That was the direct grant system abolished in the 70s and since then those schools have had to raise the funds themselves to continue to at least attempt to offer the same opportunities.