I learned all my grammar by doing Irish all through school. The only other language where I learned grammar in a systematic fashion was German, but it didn't expose me to all the ins and outs that Irish did. I agree that Latin (or Greek for that matter) would have provided a chance to teach grammar thoroughly too, but doing Irish provided that foundation. It gave me the tools I needed to get to grips with other languages later (a point Tinuviel made wrt Latin). This was also the experience of an uncle of mine, who did Irish all through school and learned French pretty easily much later for professional reasons.
The DDs have so far opted for French in high school, after wasting years doing Spanish in elementary school (in the US). Their choices there were French, Spanish, Italian, German, Japanese or Mandarin. DS took Latin mainly because he didn't have to speak but could rather focus on grammar, translation, vocabulary and historical and cultural context, and he loved it. All the DCs ended up with a solid grasp of grammar thanks to English class in school in the US, beginning at around 3rd grade (age 8ish) and continuing through high school, so the deficiencies of the elementary Spanish curriculum didn't bother me too much. Because of the way American bachelors degrees are structured, DS will take another year of Latin before he graduates, as proficiency in a language is necessary. If he ends up doing medicine then Latin may even be useful to some extent in pharmacy and anatomy. ExH did his BA in Latin with Ancient Greek as a minor, in an American university.
If Classics (and I include Irish as a classical language since literature in Irish is almost as old as Latin and Greek literature) are taught properly I think they can go far to help decode modern visual culture and certainly modern literature and drama. DS found his high school Latin helped enormously in some of his liberal arts core courses (the American system requires more of an all-round education even for those on science tracks). Many of the themes we find cropping up again and again come straight from the classical imagination.
I have the impression that the British take on education is quite utilitarian as opposed to other European countries that I know about from personal experience where being well-versed in liberal subjects is a sign of being a generally well-educated person and it is not all a matter of "what job do you want to do?" which seems to be the main interest of parents in the UK. I do agree that education will need to prepare dc for the world of work but that is not the only role of education IMO. Actually I think this is the main reason that so many traditional subjects have fallen by the wayside which is not the case in other countries.
I agree with you OP, and I think the American system values the liberal education just as much as other European countries (outside of the UK) do. Education should produce flexible minds that are capable of making connections among many different areas; I think what the British system does is produce very inflexible minds that do not adapt and that can't draw on a wide knowledge from a broad range of disciplines because of the early specialisation of the A level system. The world of work and innovation requires cross fertilisation of ideas and multi-cultural fluency.