The tactics of the two breakouts are very similar: some people set up a diversion or two (the dragon firework; Peeves with dungbombs) to attract school attention while other people break into U's office, free the prisoner(s), and cause damage.

So once he gets "a tactical team of sixteen-year-olds exists, and I had a run-in with them at the DoM", as soon as he connects that to Harry he's got "a tactical team of sixteen-year-olds exists and is at Harry's command." From there it's hardly a leap to "and they broke him out of Umbridge's", and from there it's only a moment to analyse other actions.

The big leap is not Fawcett, it's Harry.