It's not fair to say if a child doesn't pass the 11+ that they wouldn't get an academic scholarship.
Some 11+ tests are used to rank candidates. The people who get one point lower than those who squeeze in, are just unlucky really.
DD1 is at a superselective like that. She was offered a scholarship at her old school. One of her best friends was offered the same scholarship, but did not get into the grammar.
But most scholarships are tiny, as people have said up thread. You need to ask about bursaries. And bursaries usually have to be applied for each year, so you need to be on the ball to have all your financial information ready.
Different schools calculate bursaries differently. For example, at DS's school, they would take assets into account. That is, if you live in an expensive house, they would expect you to sell it or release equity in it, rather than grant you a bursary. I know that others don't do this, and take only income into consideration.
(I know this because I know someone who plays this bursary game every year, and has had to do so since her husband died when her children were in junior school. She has had to tout her children round to the various schools, comparing offers of scholarships and bursaries to work out what the best deal is. It may not be something you fancy doing. I wouldn't like it, but if DH died, I would obviously have to consider what was best for my children, just as she has had to.)