First off, material softness / hardness has nothing to do with stiffness / flexibility, which has nothing to do with strength / weakness. Steel is 3 times as stiff as aluminum (not 2 X as you stated), because its modulus of elasticity (Young's modulus) is 3 times as much. This is the same for all grades (strengths) of steel or aluminum. Steel is also 3 times as heavy per volume as aluminum. The bending stiffness of a thin walled tube is linearly proportional to the thickness (t^1) and proportional to the cube of the diameter (D^3). The weight is linearly proportional to both the thickness and diameter. Therefore, it is theoretically possible to make a tube out of either material, which has the same weight and stiffness. If the diameters are the same and the steel tube is one third as thick, then they will have the same stiffnesss and weight. Oh, and for those of you who wish it bring up the D/t limit for local shell buckling, the D/t limit for steel is also 3 times that of aluminum, so it is theoretically possible to make a steel tube that is one third as thick as aluminum, without reaching the critical load for buckling. The proneness to denting of a steel tube with a thickness that is one third that of an aluminum tube with the same diameter is the same. From a practical standpoint, however, there is a limit to reliable drawing of thin walled tubing, which is why you don't see oversized steel tubing with super thin walls.

My point was more than since Al is lighter but less stiff/softer than Al, so you can make big tubes with thicker(compared to steel) walls that are stiff and light. A large diameter Al tube with thinner walls is going to be lighter than a small diameter AL tube with thicker tubes to make the bar have the same thickness.

If you need large tube bars or not is debatable until you hit clydsdale status IMO.

First, you are mixing terminology incorrectly. Stop talking about softness; it has nothing to do with this discussion. Second, you seem to be implying that aluminum somehow has an advantage over steel, which it does not as I explained previously. You can create oversize steel tubing that is comparably stiffer and lighter, just as you can with aluminum, because the ratio of elastic modulus to unit weight is the same. In fact, you can do this with titanium as well, although the ratio is not quite as favorable. So, the only part of your point that is correct is that a tube made out of any material will be have an improvement in stiffness to weight if it is made with a larger diameter while reducing its thickness. Also, if the increased diameter is only in the clamping area, and the remainder of the bar is a reduced diameter, typically 24mm, then the overall stiffness of the bar will be substantially the same. This particular bar will probably be much stiffer, not so much because its size (diameter) has increased, but because it maintains the diameter for its entire width.

Both bars pictured in the thread are more or less straight tapered until they transition to the drops, so for they are going to be lighter for the same stiffness compared to a smaller diameter bar, which is the point of being made so large. I think my Shimano Pro 7s bars are around 250g, which is pretty light for an 80 dollar handlebar that stiff. I just got them to fit my monster hands though.