Does the usefulness of the concept of a social welfare function stand or fall on its mathematical properties?
And I answer:
The concept of a "social welfare function" (with or without mathematical properties) is meaningless. You can write equations until kingdom come, but no equation you write can make commensurate the happiness or unhappiness of individuals.
Consider the case in which a nation (call it US) is formed in order to defend its citizens from outside attack by an enemy nation (call it AQ). (That's the main reason the United States was formed, strange as it may now seem.) Assuming that the citizens of US are unanimous in their opposition to AQ, and unanimous in their support of measures taken to deter AQ, each of them will be happier if their unified support actually deters an attack by AQ. But AQ will be unhappy (or less happy) because it can't attack US with impunity. The happiness of US (even if it could be expressed mathematically), isn't offset by the unhappiness of AQ (even if it could be expressed mathematically). In fact, US's happiness is increased by AQ's unhappiness, even though neither can be quantified.
Suppose, however, that a faction of US citizens (call it LW) is unhappy because of certain actions being taken to prevent an attack by AQ. The actions that make LW unhappy don't make me unhappy. In fact, they add to my happiness because I despise LW; anything that makes LW unhappy makes me happier. Thus, I'll continue to be happy, despite LW's unhappiness, unless and until (a) LW's unhappiness leads to a political decision to stop defending US against AQ or (b) AQ attacks US successfully.
I could go on, but I think you get the idea. My happiness (or unhappiness) is mine, and yours is yours. The best we can say is that voluntary exchange in free markets, protected by strict enforcement of laws against force and fraud, would make almost everyone happier -- and wealthier. So much wealthier that there'd be plenty of money with which to buy off the free-loaders. But that's another story.(See also this post.)