I think that I'm deferring to the judgement of the father, and not the unreliable narrator, who expressly confesses that there might not have been any alligators around whatsoever.

Regardless, even though you quoted out of context, it's perhaps worth backpeddling a bit, because even though I stated that they're definitely not going to die while riding a bike, even that is an overstatement. It's clearly not impossible to die on one's maiden bike ride, though it is still very, very unlikely.

The bicycle was an analogy for diving into the water, wasn't it? I read it as it being said that in both cases you definitely won't die.

It was, but more importantly, a kid riding their bike for the first time without training wheels can be a terrifying experience, because their brains (as well as ours) aren't very good at risk assessment.

Pushing a kid on a bike without training wheels for the first time could very easily be interpreted as attempted murder by a precocious child. That doesn't mean that their fears are rational. Swimming in a river has a non-zero chance of alligator attack, statistically, alligator attacks are very, very rare, and you're far more likely to die of regular old drowning than by alligator.

They almost never attack except extremely young children.

Crossing a road or taking a car ride is likely more dangerous.