1st off Lady - wow. Respect!
On to the question specifically of why some parents will aim for the top schools for their kids - it's simply because they can, without it being too much of a stretch to them!
If you're not, it's (among other things) because it's too much of a stretch, based on your wealth, your child's ability, your level of organisation, where you live, your current social circle, etc.
OP - CAN you? Actually? Get your child into St Paul's? Or is this hypothetical?
Like many people have said here, they want the best for their child that they can reasonably achieve. This very much applies to the top-tier schools as well.
I recently met a mum who sent FOUR of her kids to a Westminster type school. Wow. I was impressed.(She was lovely by the way).
If you earn 1 or 2 million a year, the cost/effort of tutoring your kids is the equivalent of buying WH Smith CPG worksheets for a person that earns 50k. (I'm talking London). My mum bought CPG worksheets for me btw.
I have ONE of my kids in a Westminster type school. Based on our income/my organisational skills / child's ability, that's where we are. (p.s. I feel surprisingly extraordinarily proud of my child AND myself for acieving this). If all my kids could have gotten into St Paul's - why NOT? Why NOT aim high? For fear of disappointment? If so, i'd speak to a therapist about emotional management.
Also, like others have said - it doesn't GUARANTEE financial success. But it does increase their chances of financial success (and Confidence). Yes, you could spend all that money/ effort and the kid.... earns 15k a year. Yes, I would be disappointed. But we'd take it from there, and hope they made the very best of that 15k. Does that mean I shouldn't have tried? They COULD also become head of....Barclays. Tesco. Bp.
I had a 19 year old nanny recently, who said she wished her parents had put in the amount of effort that I put into my kids (or at least had made more of an effort) to secure a better future for her. I thought that was a nice thing to say.