It easy to see that your strategy should be switching if you look at this way. If you don’t get the prize on the first pick (with a probability of 2/3) you automatically get it after you switch, but you don’t get after you switch if your first pick was the prize (with a probability of 1/3). Thus by switching, your conditional probability of getting the prize is being weighted towards the greater probability on the first pick, that is the probability of not picking the prize in the first place.
Excellent summary!
By the way, back in the “Ask Marilyn” days when this was a hot topic, several classrooms of kids physically tested, through a classroom game show demonstration, the assertion that you should always switch and sure enough that answer was supported by the class’s empyrical data.