Why do aqueous solutions always have to be electrically neutral?
Because the electrostatic force is much stronger than you can imagine. Yes, you can get charge-rich areas in a cell. However, the amount of charge held here will be negligible. A rather small amount of charge can overcome the field near the plates and prevent more charge from approaching. The amount is quite small; it's not something that you would notice (and it won't give you a static shock1

1. Everyday static shocks are stronger by orders of magnitude because they deal with a lot of charge accumulated on the surface of an insulator, while here it is a relatively small amount of charge dispersed through the volume of a conductor