Scholarships are less common than they were, say, 10 years ago. Even then, they were in the region of 10/20%, so didn't really help someone who wasn't already able to attend.
With work towards verifying compliance with Charity Commission fuzziness, most schools have transferred their scholarship fund to the bursary fund.
A well run school will not have a vague, ready to expand, bursary fund. They will have strict rules about how many bursaries and who qualifies. If you don't qualify, then you don't get. If a school does roll over when you beg, then that is something to worry about, in terms of the overall finances of the school.
The vast majority of bursary funds are carved out of existing fee income. Very few schools have the endowments of Eton, for example. Most parents struggle to pay full fees, so it's unreasonable to keep asking them to dip into their pockets again and again. They are the same parents who are the squeezed middle, expected to pay higher taxes, lose child benefit etc.
The bursary fund cannot be spread across all those who are offered places. It would not mean a low-income family would be any closer to being able to attend. The remission has to be very significant, eg 80 - 100%. If they have decided that two or three other candidates are more worthy of benefiting from the bursary fund, you have to just accept that.