>In most European countries, people will know a tiny bit of broken English at least, and most of the words you can figure out on packages/signs by their roots.

I mean, what do you expect? European languages are all related to each other, so of course it's easier to figure things out by roots. Japanese isn't even remotely related to English, except through borrow-words. That said, I'm usually able to figure a lot of things out by simply reading the katakana, since 95% of the time it's just a borrowed English word. Of course, this depends a lot on what you're looking at: western foods almost always use katakana borrow-words, Asian foods almost never.