I don't think it is possible to construct a useful theory with the absolut square of psi (or its
square) as variable: psi as a variable allows for gauge freedom - and the gauge mechanism describes the way external fields act on the objects described by psi. But it is a good
question
Schrödingers equation is called a wave equation partly for historical reasons
(Schrödinger thought the equation he found would describe real waves) and
partly because it admits wave-like solutions in many situations. But the meaning
of these wave-like solutions is different from real waves...