HAI is a good school and although busy, you are still getting the quality training that it is renowned for. I dont know of any other school that can match everything they do, and in terms of hours, the people that say they only get 2 hours a week are probably not pushed to fly much more than that. If you are hard working and push hard with your instructor you can get everything done in less than 9 months, and then you have 15 months to work to build as many hours as you can. And since you all know that HAI is really busy, then think of it this way. If you get hired there you know you will be flying your ass off with students. If you dont get work there then the rest is up to you.
For visa purposes, the best way to get the most of your time is ask for an M1, come over and do your private and instrument and most of your commercial, then go home and get a J1 visa, comeback sit your Commercial checkride and get your CFI and CFII and then you should have loads of time to build a substantial amount of hours while working.
If you have the money I would advise doing the FAA and JAA together. It will require you to put the work in and not sit around on your ass, but having both will pay off for you in the end.
Good luck with the flying. And dont forget, HAI is situated in the middle of Florida. So compare the weather there to Oregon and other places and tell me who you think will get the most flying year round. Common sense really!!