It's a myth in the most literal way. Fleming published and promoted his results despite a lack of reproducibility. By the time he won the Nobel Prize, he had backformed or misremembered a folksy story about an open window. That's textbook mythmaking.

It can both be fine to have a glib story to tell schoolkids and important to recognize that the actual intellectual process is messier and more complex.

The path to his discovery may have been difficult to replicate, but the fact that the mold could kill other bacteria was not, and was immediately replicated.

It just wasn't seen as relevant because, at the time, few people imagined its internal use in humans and it was instead seen more as a tool for other microbiologists and the like. The jump to "And then I see the disinfectant, where it knocks it out in a minute. One minute. And is there a way we can do something like that, by injection inside or almost a cleaning?" took quite some time.