I don't object to paying twice, in the sense of paying tax to support state schools that I don't use and then paying fees. As stated, that's normal: taxes support all kinds of public services which we as a society need to be there, whether we use them or not. In fact I would like state schools to be much better funded and I have always voted accordingly. I think education is very, very important and that underlies both my personal and my political actions.
(Where I disagree with some here is when they assert that by not using the state system I damage it, even, damage it enough to offset both (a) the money I save the taxpayer by not using it and (b) the good that is done to my child and to society by his being educated as well as I can arrange. Those seem to me to be extraordinary claims that would require extraordinary evidence.)
The part I do strenuously object to is the third time I pay: the way my fees have to be used not only to pay the costs of the independent school, but also, to pay the costs of the independent school supporting local state schools. Fine where this can be done at a marginal cost of zero (allowing free use of playing fields at times when they're not required by the independent school is probably the commonest example) or where there's a mutual benefit that's large compared to the cost (many joint events). On the other hand, stuff like, people who are employed to manage the independent school being expected to spend time, via sponsoring an academy, managing a state school, or teachers being expected to mark work of visiting state school pupils - those are using hours our fees are paying for to subsidise state education. I'm already paying what someone in my financial position is supposed to pay towards state schools. I don't expect another slice taken towards that from my post-tax spending, any more than you expect to see "State school dinner levy" on your restaurant bill or "State school field trip contribution" on your holiday bill. If - as is likely I think - it's right for people on my income to pay a bit more towards state schools, let us ALL do so.