June is indeed the start of hurricane season, but even then it's still a little early for Grand Cayman. We spent our honeymoon there in late June 2001, and have been back in June almost every other year since then, and haven't had any troubles. I've never been to Aruba, so I can't compare - but we fell in love with Grand Cayman the first time and have not seen the need to go anywhere else.
The RC is enjoyable - we stayed there last year, but we find that we prefer the quiet of the East End so we stay out that way most of the time. I didn't have the Amex Plat yet last year, but I believe that you should be able to find a reasonable FHR rate at the RC.
You will not find yourself lacking for fine restaurants, but I'd suggest that to find a couple of our favorites, you'll need a car. You can check out
this post of mine to find out about one restaurant we make sure to visit at least once every time we're there. There is also a wonderful, wonderful dining experience to be had with
Chef Roland - a "pay what you feel the meal is worth" place that's actually housed in Roland's backyard, which might seem quirky, but it's truly one of our favorites. Roland is a classically trained chef from Germany, and puts together combinations of hearty European and light island dishes that are truly fantastic.
You've come to the right forum - TomCayman has a wealth of knowledge and should be able to help you understand all of the opportunities that GCM offers!