Being bilingual, English-Danish, I agree entirely, Monica. I have never been able to see that an activity designated as "cosy" in English is not exactly the same as anything called "hyggelig" in Danish. (But people who only speak Danish, don't agree on this point.)
Both the adjective "hyggelig" and the noun "hygge" do have a wider semantic field than "cosiness" as you can use them to describe people, which of course would be quite impossible in English with the word cosy. If you describe a person in Danish as " en hyggelig person" you are faced when translating into English with deciding whether hyggelig here corresponds to "nice", "easy-going" "easy to get on with", or less aptly, "pleasant" a word for which we have a direct translation in Danish and don't need hyggelig, although the modern tendency seems to be to replace both pleasant (rar) and likeable (elskværdig) with hyggelig.