I'm Scots, educated in Scotland, and I felt exactly the same way about having compulsory CoS worship foisted upon me. Protests were ignored, I was threatened with punishment for asking to be excused.
I am similarly not a Christian, never have been, I don't identify in any way with Christian practice or culture, so I always felt aggrieved at being regarded as Christian by default. I felt exactly the same as you with regard to "us", as hearing that only ever made me feel "but I am not a Christian!".
As for the documented positives of communal signing; well that might be relevant on a macro scale, but unfortunately if it's compulsory you also immediately scoop up all the children like me. All it did was make me angry and miserable at the time, and resentful for the remainder of my life. The notion of it being necessary to facilitate "communal singing" at public events just makes me laugh, as that's the one thing that would guarantee I don't attend.
Also a point about this "Christian culture" tripe. Yes, it's undoubtedly the case that Christianity is interwoven in UK laws, art etc, but this is yet another example of how Christianity attempts to monopolise everything. Law, "culture", art, morality all predate Christianity. What is now the UK did not have a total absence of these things prior to the arrival of Christianity. Christianity co-opted them for the most part and elbowed a lot of the pre-existing culture out of the way.
As for the laughable comment about "Do you not want children to know why we celebrate Christmas?" Well, for a start, some of "us" don't celebrate Christmas. I suspect there is no interest though in explaining why "Christmas" is just a continuation of Pagan tradition that Christianity co-opted and muscled out of the way. For a more concise answer, no, I don't believe the religious Christian elements are particularly relevant. If children are not Christian (and there is no such thing as a religious child, just children who have been immersed in religion by adults),then all they need to know is that it's rooted in pagan tradition and some people still observe it.