SCHOLARSHIPS are meant to reward the most talented kids regardless of their background... so emphasis on "scholar".
BURSARIES are for bright kids from lower income families.
Both serve different purposes.
Scholarships are more of a "prestige" thing. If you are a scholar, you are recognized as one of the best (at your school) in whatever you got your scholarship in. You are the "benchmark" for all the other kids. It looks good on CVs, university applications, etc. Now, you can be the best, despite - maybe - being rich. Why should someone who happens to be wealthy not be rewarded for also being good on his/her own account?
Bursaries are for those who may not have gotten a scholarship (though sometimes, they do get both), but in a good year with loads of talented applicants, they were probably close to getting one. One thing you have to remember is: BURSARIES ARE NOT ENTIRELY ALTRUISTIC. By selecting the best people out of those who may not otherwise be able to afford it, you lift the average which is good for the league tables, of course.
The ones who actually benefit from the above set-up without any of them getting anything "out of the pot" are the wealthier kids who may have been anywhere between "average" to "just as intelligent as the bursary kids". They are now surrounded by motivated and intelligent children who are at the school for a purpose, and know how lucky they are. Obviously, this changes the dynamic a little.
An exclusive school for only wealthy kids does not usually yield very good results: (1) Just because you're rich, doesn't mean you're intelligent; and (2) such kids can often take their posh school for granted, in the absence of those kids who work hard to be there.
At the same time, handing out bursaries to anyone who is not necessarily much better than average, just because they are low income, would not yield great results either. It definitely would not make the wealthy but average kids work harder or aim any higher, because they'd be surrounded by people who are actually just the same as them.
The whole scholarship/bursary thing is a give and take. I guess in years that aren't so great (i.e. less talent), there will be more people of "average" ability who will get them. But in recent years, where there is less money in general, but a large pool to choose from, it will be difficult getting a lot of aid unless your child is super-bright.