Pretty much. Some LEAs maintain one each of single-sex Comprehensives for parents who might have issues with co-education. The same applies to the LEA-funded Grammar Schools. Church Schools, which are also LEA-funded but provide an education and culture built around their faith, can also choose to be single-sex, but quite generally the only ones that take advantage of this are the Islamic schools (note that Islamic Girls Schools do not usually provide education beyond the minimum school-leaving age of 16, but the Boys Schools do.) Thus yes, the majority of single-sex schools are private ones.
As to what I propose, that takes us into different territory!
I would break up the huge Comprehensives into a set of smaller schools, each with their own character and culture. Thus some might place more emphasis on sports, others might stress strict discipline and formality, there might be some which encourage STEM subjects whilst others veer towards Liberal Arts. All would teach the same basic curriculum, but in different styles and with different methodologies. Church or religiously-affiliated schools would be banned. The aim would be to give all families in every catchment area as broad a choice as possible to enable them to find the best fit for their children. The 'default setting', if you will, would be single-sex for the 11-18 age group, but that said, if a school could demonstrate that its specific culture or ethos required and properly supported co-education, then fair enough, and the parents would decide.
Now in so far as my experience is accurate, it seems that these vast, anonymous, thousand-plus-student schools lie at the root of declining academic and behavioural standards. Put simply, teachers cannot keep an eye on so many kids at once, cannot give individual pupils the attention they need and cannot protect the prey from the predators. On the other hand, a school with a student body in the low hundreds at most has a chance to develop its own culture and ethos, and the staff can actually get to know who's who, including the 'usual suspects'.
In the matter of LGBTQ pupils, it is to be hoped that all schools would be properly supportive of these youngsters, but it is also inevitable that some would provide a culture better suited to their needs, and again, parents would have the choice of sending their kids to those schools.
In closing, I will say that the argument for co-education was (and is), that by working and learning together, boys and girls will develop understanding leading to mutual respect. The argument against was that 'familiarity breeds contempt'. The current culture of systemic sexual harassement and abuse in far too many schools points to the latter being correct. That said, the current over-emphasis on sex in our culture is also at least partially to blame.
Pie in the sky? Probably, but you did ask!
