My understanding/head canon is that the entrance exam Wesley takes would be one specifically designed for super geniuses. He had already been labeled "space Mozart", and the other candidates are also portrayed as high achievers, if not outright savants.
Why a scientific exploration force wouldn't accept all suited applicants before the Daystrom Institute incubator program snags them up is beyond me... What we see in the episode seems counterintuitive, and more like a military approach to selecting "the best of the best" through competitive testing. I understand your confusion π
But the Doylean reason is of course that they needed Wesley in a sharply defined dramatic trial situation, and could neither afford nor coordinate 150 extras to portray a more "realistic" entrance exam.