Actually, for center-of-gravity reasons they (often) cannot reassign seats freely, as the aircraft gets balanced based on the assigned seats. If everyone moves up front, the pilots might get a surprise in how the plane reacts.
It is not about balance.
Moving a few people probably doesn't cause an issue if the rest of the plane is completely empty.
If you move 20 people from the right side to the left side of the plane this will probably have some kind of effect.
On smaller regional aircraft they sometimes ask people to move seats to the other side or back to front.