I think you did a great job making a distinction between the United Kingdom and the separate, distinct countries which comprise Great Britain. A lot of your points are well-taken, and your examples are spot on.

The one area that stuck for me was the following:

"While England is the largest of the countries on the island of Great Britain, and has the largest population, it is not another word for the UK. It would be like referring to the USA as New York, or Texas. The Queen is the Queen of Great Britain and Northern Ireland, not England."

I spent quite a lot of time thinking through whether that was accurate, and while I think it ultimately is, the imperfection of the analogy definitely threw me off. I wonder if a non-governmental example would be a better analogy so you don't get readers hung up like I did on whether, for example, referring to the USA as one of its states, or whether the monarch of a commonwealth is also the monarch of its collection of smaller countries.

Overall, I think this was an effective, persuasive piece of writing. Well done!