There has to be more than one academy In the show and a shit load of fics it shows that their is only one academy to train ninja. But the problem stems from the fact that there are literally thousands of Shinobi in Konoha and the system for how the academy is structured makes it even harder for it to make sense.

Like the system seems to be, one academy, one class. All these classes are separated by age, so there’s no other form class where the age group is the same. That means only a handful of Shinobi come out of the academy every year and that handful becomes smaller since most Genin fail their exam.

Konoha is a literal superpower yet only turn out handfuls of Shinobi a year, that makes no sense story wise and there should be more academies in the village or the academy has to be reworked