I guess you really know the answer to some if your questions - why do so many apply - well because like you, they think fee paying schools will be better for their kids. Remember, the figures are deceptive though and when 10 apply per place, if they've each applied to 5, but can only take 1 place, the odds of getting a place are not as bad as they first look and most schools significantly over offer because even the very good ones won't be top choice for every applicant.
It's often said that being top table doesn't tell you a whole lot. Primary schools vary a lot and getting to the top table can mean a student is exceptionally clever, or amongst some cohorts, would be below average nationally.
You just can't always tell how they will do in exams. Some super bright kids aren't that well prepped for doing the exams themselves with things like timing, knowing exactly what's required and so don't impress. Others just struggle under exam conditions and don't reflect their true ability. Bright children miss out every year.
Good advice is not to apply to somewhere you aren't interested in. So you got your 3rd choice and it wasn't top, but presumably you were prepared for your child to go there or wouldn't have applied and paid the entrance fee. Some kids get no offers but you've got one that presumably you did like.
In the end its demand and supply. There were other applicants that met the schools criteria more strongly. It is a competitive system and in the top schools there are finite places and far more applicants, so children have to accept they might not get their aspirational place but need to take one that wasn't quite so high on their list. Usually people go onto love the school they go to and can see it suited their child.
Is no good feeling annoyed at the system...you chose to enter it, knowing the risks and that the odds are not in your favour....you mentioned the statistics. Of course not everyone gets places in the very top or the next tier or even the next tier schools.
The key thing is that your child doesn't sense your disappointment or annoyance at the system....that just sets him up to be cross and not pleased with the school he will go to. It's all about building the excitement for the next phase now. Work on the basis the wait list won't come off but build up the offer he has. If the wait list does offer a place, it's an added bonus....you can wait anxiously to hear, but he shouldn't have to do that, but have a sense that he's done well...not that working hard hasn't paid off. It's a mistake to feel like that...without the hard work, perhaps he wouldn't have any offers. No one deserves a place at a top school just because of hard work...it doesn't work like that - they want the cleverest, not the hardest working kids. What a shame if he is feeling strongly discouraged now. Parents are so important in helping their children have a positive attitude all through the process...to know that all the schools you're applying for are good ones, that they do t need to worry because thee will be a good school for them, that you're proud of their efforts and pleased about where they will go to school when the offers come in. It's so important because no-one can be sure if where their child will end up, and starting somewhere feeling its second rate or beneath you as a family or child just isn't a great start.