We have studied experimentally the states formed in artificial square ice nanomagnet systems following demagnetization in a rotating in-plane applied magnetic field that reduces to zero in a manner that is linear in time. The final states are found to be controlled via the system's lattice constant, which determines the strength of the magnetostatic interactions between the elements, as well as the field ramping rate. We understand these effects as a requirement that the system undergoes a sufficiently large number of active rotations within the critical field window in which elements may be reversed, such that the interactions are allowed to locally exert their influence if the ground state is to be approached. On the other hand, if quenched disorder is too strong when compared to the interaction strength, any close approach to the ground state is impossible. These results show that it is not necessary for there to be any ac component to the field amplitude that is applied to the system during demagnetization, which is the method almost exclusively employed in field protocols reported to date. Furthermore, by optimizing the parameters of our linear demagnetization protocol, the largest field-generated ground state domains yet reported are found.