Oh come on, it is a bit naive to say that lesbian as gay doesn't have an impact on Lesbians.
It may not have much impact on a British person living in Britain, but I can imagine that it's pretty annoying for a Lesbian man to be greeted by sniggers every time he says where he's from. Or a Lesbian woman having people assuming she's making a statement about her sexuality. And so on.
Another thing that I think you haven't considered, is that when you don't live in your place of birth, you find yourself explaining your nationality a lot. I probably say "I'm English" or "I'm from Oxford" at least once a week to strangers, because I don't live in England. Therefore, the word becomes more important.
thirdly, we are not used to controlling English. France and Germany have academies that can, if they so choose, direct the language far more precisely. For example, Le computer was "out there" but I have noticed that the people Ive come across now seem to say l'ordinateur instead.
Taking these things into account, the islanders' campaign doesnt seem so unreasonable.