I understand your friend's logic. And I have the 10-24 and like it. The ability to go wider is important -- you can always crop narrower afterwards, but if you can't fit the whole subject when shooting, it's lost.
However - you are clearly asking about "when you only have a single lens", and the 16-80 does sound better for that (I've never had it). For what you say you want, I think the 10-24 plus the glass you already have isn't much of an option.
I do have the 18-135 and I really like it. There's so much it can do. I guess it's pretty big and heavy. It may or may not be worth your consideration. Again, you can always crop -- maybe travel photos don't mind cropping?
