I'm actually finding it really fascinating reading people's different experiences and likes/dislikes.
I've only done AI once and that was over 20 years ago! We did a last minute AI to Dominican Republic because I had just had notice of redundancy so we wanted something where we knew what we were spending. We stayed in Puerto Plata region. Can't remember the resort name and no doubt it's gone now anyway.
I didn't have kids then so probably didn't take any notice but I don't remember any activities for kids, in fact don't remember seeing hardly any kids at all. It was basically breakfast, swim in the pool/use loungers, lunch, swim on the beach/use loungers, dinner. All accompanied by lots of booze. Plus our room had a TV which received some US satellite channels, which was a massive novelty. We spent a fair amount of time watching infomercials!
Since then I spent about 10 years doing hotel/package to various places, but did a self catering villa for the first time about 10 yrs ago and would never want to do anything different again. Own pool, washing machine, can cook own food, no noisy neighbours, no falling into the trap of "eat and drink as much as you can, you paid for it!"
If you have teens then just pick carefully - many villa owners now provide wifi so you can take a games console. Make sure you're near somewhere that has some decent activities eg water park.
One of the things I love most about travelling is eating local food, and I don't mean "Mickey's Sports Bar, Full English Served Daily". Unfortunately I just don't think you can get this with AI. You certainly won't get the experience of something like a mezze.
My DC has issues with IBS and really struggled with unfamiliar foods but I've been able to encourage him to try different stuff when we're away and it's really helped him expand his menu. I'll often recreate meals at home that we've had on holiday.
A divisive factor which I've seen through this thread is that some people want to go on holiday and meet/socialise with others. Whereas others (and I'm in this camp) want to just keep themselves to themselves and enjoy some family alone time.