I say the same to everyone who asks these kind of questions, if you have a great state Grammar option, then, unless you are loaded and the £150k fees are really a drop in the ocean, it makes sense to pick it. The Independent option might offer some extras and a different experience, but it's probably not worth £150k if the state alternative is also very very good.
The answer might be different if the state alternative were a rubbish option - but it's not in this case.
Re the tutoring thing - it's widespread for entry to both of these schools. When people can afford fees, perhaps their approach to tutoring and preparation is understandably not as intense - because they know there will be other good options if the very top choice doesn't come off. For many who can't pay fees, the alternative to the state grammar seems very poor and that obviously is a big motivator.
One final thing - many of those who get offers for independents don't get offers from the super selective state grammars, if they sit both. Many also choose not to sit for the state grammars, saying that they don't like the tutoring ethos or very exam focused education of offer. Some do enter, but say that they are only doing it for practice and not prepping for it and wouldn't take it if offered - strikes me as a way of covering yourself for when no offer comes.
I think that it is a simple fact that it is more competitive to get a state grammar place if it superselective. Yes they have often been tutored, but the kids getting independent results have often been tutored too or had years of Prep schooling or both. In most cases, the superselective state grammar will have a less wide range of ability....less of a relative tail.
The independents though, do manage to get some fantastic results out of their wider ability range. So my observation for parents with money, is that if your child is bright but not super bright and likely to make the state superselective cut, they may get an independent school place in a great school and end up with similar results.....so for me, it's those kids who really gain from having the choice that money gives. The kids who are bright but not super bright whose parents cannot pay, might just have to take their chances in the local Comp.