I adore Fire Island. A week spent there last year in a house with friends was the most relaxing, social, revitalizing vacation I've had in years.

But I would not have said it meets any of the OP's criteria. I can't recall seeing a single child, nor does it have good hotels - notoriously FIPPOA makes it hard to open one and those that exist are poor (although I think there may be one decent newish one) - and the only way to do Fire Island properly is to share a house. You also can't drive there - which is one of its joys.

Perhaps there is a different part of Fire Island to which you're referring. But if you just type it into Google - or Google images - it's pretty clear that to most people "Fire Island" means the part I know.