To the point of "That person isn't using it as a simple weapon" that has zero meaning in 5e. A person is proficient or not in a weapon. There is no double, triple training in weapons like in 2e. A wizard using a staff is just as good at fighting with it as a fighter is. The fighter just knows how to make more attacks, and likely has more strength to put behind the swing. But the proficiency bonus to hit is the exact same.
I think the real issue is when comparing it to a club, people think of a club as closer to a mace, when a club is a light weapon. Basically in 5e, its just a bit more than a handle.
Also I think its absurd to assume that just because you are holding a shield, the hand is completely incapable of assisting with the other hand in combat. Obviously you are going to be using two hands on the weapon at all times, but it can brace for moments when momentum or such need to be adjusted. The shield is strapped to your arm and will not fall off from a momentary adjustment. Even bracing or moving the weapon with the shield itself is fine.