My DS1 came top in the entrance exam for his competitive entry independent senior school. He was already in the prep of the school at the time, though joined in yr 6 from a state school so had only been there for just over a term when he sat the exam with no additional preparation. He wasn't just one of the top set at primary school though - he was miles ahead of the other children academically. DS2 was a solid top set high achieving child, but didn't come anywhere near getting an academic scholarship.
The scholarship DS1 won is worth £900 per year - it does not make much of a dint in the fees, though it is very nice to have. If you couldn't afford to pay I doubt it would make the difference IYSWIM.
There are bursaries to DS1 school which are worth 25, 50 and 75% off the fees but they are based on scrutiny of parental income and are reserved for people on lower incomes than yours I would think.
DS1 came top in the exam for another local independent school and was offered a free place - we would just have had to pay for compulsory school dinners and public exam fees (which they billed separately). We didn't take up the place as it was not a competitive entry school and decided it was not right for him. This school was not a charity - I don't think a school could offer a free place which did not take account of parental income and hope to remain a charity.
People at work often make comments about how I can afford to pay school fees when they can't. I think the answer is in things like I am driving a 6 yr old car, DH is driving a 10 yr old car, we don't do expensive holidays etc whilst my work mates tend to have far superior holidays, cars, nights out etc. I would think on an income of 75K and no mortgage it would be doable if it is your top priority - we do have a bigger income but also a big mortage and three children rather than two to pay for.
My DS2 actually travels about 20 miles to a grammar school in another county for which he sat the 11+. Hardly anyone near us had ever heard of it when he sat the exam, even though it is not that far away. It might be worth researching to see if this option is available to you in a neighbouring LA?