Complex sentence: It is important that the properties associated with the wave function are "invariant under normalization", otherwise we still cannot yield any information about the particle associated with the non-normalized wave function.
Split into simpler sentences: It is important that the properties associated with the wave function are "invariant under normalization". If normalization of a wave function changed the properties, the process becomes pointless as we still cannot yield any information about the particle associated with the non-normalized wave function.