I was relatively late to lose the stabilisers, and certainly didn't get the hang of it until after my younger brother did. This was largely through lack of confidence, but also because I had very few opportunities to ride other than round in circles in the back garden, which made it hard to get up any decent speed and taught all the wrong weight-distribution habits.
What made the difference for me was watching the Big Kids in the park, playing ride-as-fast-as-you-can-then-jump-off-and-see-how-far-the-bike-will-go. I don't remember, but my dad may have even given my own bike a good running-shove to demonstrate that it would indeed do the same thing. This provided much-needed empirical evidence (I was that sort of child) that bicycles have some sort of magic stabilising force that increases with speed, and that riding on two wheels is not, as one naively assumes, equivalent to the easily demonstrated to be practically impossible task of balancing a stationary bike.
As an adult who has recently learned (and taught a couple of other adults) to ride a recumbent, I can summarise this as "Moving bikes are self-stabilising. If in doubt, go faster.", but that might not be as helpful to a child who may not have the same pedalling skills or faith in the laws of physics. Fear of speed is probably an issue too, so perhaps stabilisers aren't entirely evil, as they let you get the braking/stopping bit sussed first. A bit of risk compensation through use of protective clothing might work to your advantage there, too.