My kids did the clubs once (Simba's Cubhouse) and my DH and I ate at Jiko (so we were right there, and we were staying at AKL). They were almost 7 and almost 9. My DD loved it, but my son, almost 9, really hated it. He was the oldest kid, and just played video games. It wasn't horrible, but expensive and he was bored. I know lots of kids like the clubs, and I wanted to give it a try, it was the holiday time and DH and I wanted a quiet dinner (and we didn't think the kids would like Jiko too much).
We've left the kids at 8 and 10 at the Boardwalk one night and went to Epcot for a couple of hours. They were much happier staying in the room, and my son had his cell phone so he could call me and vice versa. They kept the chain on the door, so really, they were fine. It really depends on how you feel with your kids, though, but that was fine for us.
In our county (I looked it up years ago on our police website) they can stay home alone at 8, and watch siblings under age 8 at age 12. Our kids have stayed home alone for short periods of time since then, so they're comfortable with that. They've never been in daycare, so they're usually with me.
At the ages the OP is mentioning, I personally find dinner with the kids fine. We don't go out too often without our kids, but my DH works a lot, so weekends he likes to spend time together with us all. We have taken one 5 night vacation away, though, which was really great!
I'd either take the kids to dinner or let them stay in the room. They'd be fine if they have cell phones, and you're right there at the resort. I'd be much more comfortable with that than hiring someone to come in, and when I was 11 I would have hated having a sitter. I don't know what a sitter would actually need to do other than be a warm body anyway.