The entire first world lives like a king. Think about it - back in the days when being a king actually meant something. If you were king (or at least high born; royalty) you lived in comfort and could have anything you wanted from the far reaches of the world. This is of course, ignoring the whole 'ruling the people' part of being king, but the term "live like a king" isn't referring to the political side of things, but the part that is being surrounded by luxury.
You go to the grocery story and pick food from around the world to make a nice meal for you and your friends. After the meal you sit around in your comfortable furniture in your house full of stuff and drink fine wine (ok, maybe a bag of cheep red, but still) with no cares about the daily struggle for survival. THAT is living like a king.
You can easily put a skiing spin on it, but why would you? It'd make it awkward and not really address the true reasons behind the saying.