cake There is no need to apologize. I do not believe that our exchange has been uncivil. I do believe that we have been talking past each other to some extent and my tenaciousness has been driven by my desire for clarity and precision.
I am a parent and, while we have applied to a GDST school in the past, my DDs are not in GDST schools nor do I expect them to be in the future. While being fortunate enough not to be eligible for bursaries myself, I am interested in the subject of how they are funded and the question of whether I am being forced to subsidize them at my DDs' school.
The Sutton Trust report to which you linked earlier highlights that some schools do indeed openly reserve some fee income for bursaries, thereby forcing full-fee paying parents to explicitly subsidize bursaries. However, other schools are able to pay for bursaries through a combination of (current) donations, investment income from endowments (i.e., past donations), and other income. It may be that at these schools parents implicitly subsidize bursaries; after all, if the school didn't offer bursaries it might be able to reduce tuition fees. However, my belief is that much or most of the donations, both current and past (the endowments) were given for the purpose of bursaries, so if a school didn't offer bursaries it wouldn't have the extra income to reduce school fees anyway.
Eton and Westminster were founded and endowed (generously) to educate "poor" boys in the days before the state provided universal education. I agree it is unfortunate that English monarchs failed to similarly endow schools to educate girls. Some solutions to that inequality might be for the boys schools to go co-ed or for them to establish "sister" schools and share their endowments, but I don't see anything like that happening soon.