So besides your history with AD&D, is there any reason your actual kids want to play that instead one of the thousand other games in various genres out there?
And while with an AD&D background you may find some of the FATE mechanics new and odd, for someone just starting in the game, do you have any reason to think they'd have more trouble picking that up rather than the arcane rules of AD&D?
I have an eleven year old daughter. I got her the new D&D Red Box in a fit of "wanting to get her into gaming" since she'd expressed some interest. She didn't open it.
She was really into Doctor Who, so I got her the new Doctor Who: Adventures In Time And Space game. She read it and enjoyed it, but never really wanted to play it - still too much, too complicated.
So I got her Heroine, a much smaller girl-oriented storytelling game (58 pages), and that did the trick.
A friend has a ten year old son. He got him and his friends into gaming via the Pathfinder beginner box and the somewhat silly goblin modules from Paizo.
All kids are different. Figure out what kind of genre/content they are interested in, and how much rules they are likely to bother with. It's cool to GM for your kids, but if they're not rapidly empowered to play with each other without adult help they're not going to take to it. Eleven is way past the "hang on everything the parent-heroes do" and is moving quickly into "anything parents are into is probably lame and will get me ostracized at school." I like Freeport and have run a lot of Freeport. Is "pirates" what your kids are the most attracted to, or is it something else?
There are thousands of RPGs of every genre and complexity, from clones of the old games you're familiar with to new big fat games (FATE counts here, it's not light) to newfangled indie games, to the point of one that uses pulls from a Jenga tower instead of dice as the resolution method.
So from one perspective, "Kids" and "I used to play AD&D" are not really enough to narrow the field of likely good games down to less than 100 or so (and just for the record, both AD&D and FATE are probably two of the games I would least recommend picking for eleven year olds). But on the other hand, getting into gaming is more about the experience and less about the individual game; expect them to move on to other game(s) once they get the general idea down. Back in the '80's we stuck to one thing for a long time; children today are much more adept at juggling a variety of different games, gaming systems (electronic/computer-wise too), media... Don't invest too much in whatever you're starting out with. From that point of view you might as well start with AD&D if you're familiar with it yourself and still have the books. Or the Pathfinder Beginner Box and D&D Red Box are two products designed to be "starter D&D." Lighter, and leverages your experience.
I fear that any 200+ page book - which means FATE, Dungeon World (though ironically not AD&D if you just count the PH) is going to be a turn-off at that age. And again, though "you" can read and run it, that's unlikely to turn into sustainable gaming for them.