I live and work in a large private school in Cornwall. As an employee I only pay 1/3 of the school fees - but still have to fork out for uniform, school trips, latest fad (that the other kids have) and all the other bits & pieces. Without the discount we couldn't afford to send my daughter to the school but on the other hand we could have applied for a scholarship/bursary.
If you honestly think your child is bright/talented then enter them for the entrance exam - our school sit them for 11, 13 & 16 year olds (entry into 1st/3rd or 6th year). It doesn't cost anything to take the exam and depending on their results the school may offer a scholarship or bursary.
Our school has a 99% pass rate at GCSE & A level. Yes you can argue because they are picky but also because they will push the kids to perform but within that childs limits.
I went through the private school system and although I'm not in some high flying job that wasn't down to my education - that was down to making wrong choices when I left Uni.
If the school thinks your child has ability & potential whether it be academic, music or sport they will do their utmost to get your child to attend - boosts their pass rate at the end of the day. So go for it if there is one near you