Of course - I think that most parents want the best for their children 
I would, hand on heart, quite happily send my DSs to Beechen Cliff rather than put myself through financial hardship (or even stretch the finances uncomfortably). It is a great school, sons of friends are doing v well there, and it is very geared up for teaching (and feeding!) boys - the school has done a lot of research into how boys typically learn best, and has tried to tailor its approach. (This is what I have heard from friends who have sons there.)
I also think that the school is only one factor - it's not an automatic assumption that your DS will get 'spoiled and useless in a good-for-nothing state school', surely? If he is bright and well-motivated and well-supported, there is plenty of scope for him to achieve highly - he won't be the only one wanting to make the best of it, although not all of the students will be like that.
I would be very surprised if being turned down for a bursary or scholarship at this stage meant that you couldn't apply again later - at 13, for example. And it would be worth doing, as the fees are fierce. You would find yourself paying out quite a lot on top of the £12k basic fee - many (but not all) extra-curricular activities will be extra, and trips also. The trips are also quite likely to be inherently costly - trips abroad rather than in the UK, for example. I doubt that they'd be mandatory, though - I went on all of mine, but can't imagine that it was obligatory. There may well also be a fund that could provide assistance with that sort of thing. The uniform can also be more expensive than you might expect, although good second-hand uniform should be widely available.
It's quite hard to say what level of household income would be needed for private schooling - it would depend on how much your housing costs etc. are. I would just say that we have a household income of around £100k (and a fairly normal level of mortgage, I think - nowhere near the amount that the bank would have let us borrow) and don't think that's enough - but then, we have 3 DSs, so it would be a huge financial commitment (and one which, as I've said before, I am not at all convinced is worth it). We could increase our household income if I were to work full-time instead of part-time - but that isn't something that I'm prepared to do. I'd rather spend more time with them, and spend the money saved on school fees by going on great holidays and lovely trips/experiences. But that is a very personal view, and others would prioritise formal education.
Big decision - best wishes with it.