Two of my former students did exactly what you propose a couple fo weeks ago, and they returned filled with enthusiasm. They did the classroom and referral work with me here in Colorado and went to Bonaire for their checkout dives. They just decided impulsively to go on with the AOW, and they are glad they did.
PADI made a mistake when they came up with the term Advanced Open Water, because it implies that you are an advanced diver when you are done. You are not. It is only the next step in your training. That is why they call the book and the ourse "Adventures in Diving."
The advantage of doing it right away (which, by the way, is what I did) is that you start your diving experience with an instructor who will show you the best way to do things instead of floundering around on your own. To those who say you should get more experience first, I ask "Why?" Yes, experience is very beneficial, but why not get that experience after you have been given more instruction?
By the way, PADI OW requires 4 checkout dives, not 3.