Living in Korea, it's kind of hard for me to imagine the public taking a sufficient interest in European affairs to give NATO operations a sustainable degree of popularity. As I've mentioned elsewhere on this forum, when discussing World War II with Koreans, they often seem barely aware, if at all, that Hitler was allied with the Japanese, for example.
Not to single out Koreans as uniquely uninformed. You'd probably get a similar degree of "Huh?" among the Canadian population if it were suggested that they join ANZAC. (Though the British-connection might help a bit there.)