I've been twice but it was around 15 years ago so I couldn't really suggest any place to go since I don't really remember and the 2nd time we went some of the places we liked were no longer open.
If you like the approach in that video, you'll like the one at St. Maarten too. You come in over the water very close to the beach and right at the other end of the runway is a big mountain. Enjoy!
People hang out on that beach just to watch the planes come in..you may want to do that one day that you're there.
There was a restaurant to the right of the runway on the water in some resort that was open the first time we went but closed the 2nd time. It was really good and a nice setting...maybe someone else has it open now.
We went to St. Barts one day...It was like a 2 hour boat ride and they were snooty over there imo...I would not do it again.
We went to Anguilla one day ...it was like a 20 minute boat ride and went to this chickie bar on the beach and ate bbq and drank "boomerangs"...that was a great day.