One truth is that if you want to list all of the exceptions then you will end up with a large number of weapons creating a list of which is best for each and every situation.
Some players are able to keep all of that information in their head and to switch between weapons so that they always have the optimal weapon for every encounter.
I can't do that. So for me I need to know what 1 weapon will do me the most good.
And, because we are talking greensteel that means gathering all the crafting materials -- something that is not a trivial exercise for me. So I am not going to be looking at 2 or 3 weapons. I am going to look for 1 weapon that will give me the most value for my investment of time and energy.
Also, because we are talking greensteel, I am not going to worry about which weapon is better when I am at low and mid levels. I am going to look for the weapon that gives me the most bang for the buck when I am at high level with my full STR, enhancement lines, etc.
If I do that 1 weapon stands out. That is falchion.
The fact is that greataxe stands out in some specific instances. The fact is that maul stands out in some specific instances. The fact is that Radiance II instead of Mineral II stands out in some specific instances. It might even be a fact that Lightning II stands out in some specific instances (although I haven't run these numbers).
But, if I am crafting my first greensteel and it has to serve multiple purposes then falchion wins. And, unlike the difference between falchion and greataxe against a boss like Harry the falchion wins by a huge amount in general purpose use.