See what others come up with, but here's my stab at an argument.
Hopefully we can agree that the only available physical variables are the ##n## forces ##F_1,\ldots,F_n## and their positions ##x_1,\ldots,x_n##. Hopefully we can agree that under different conditions the rod accelerates clockwise or counter clockwise or stays at a constant spin rate, and it would be useful if there were some function of the forces and positions that were correspondingly positive, negative, or zero.
What do we want such a function to do?
We'd like it to be linear in the forces. This way we don't have philosophical issues if we place two coins each of weight ##W## on a lever - does that count as two forces of magnitude ##W## or one of ##2W##? If the function is linear in the forces we don't care, but if it is, say, quadratic then ##(2W)^2\neq 2W^2## and we have a while can of worms there. So we've argued that our function ought to be ##F_1f_1(x_1)+F_2f_2(x_2)+\ldots+F_nf_n(x_n)##, where the functions ##f_1,\ldots,f_n## are not known and may differ.
We want our function to be zero when we have equal and opposite forces applied at the same point. For this, we need the same function of distance associated with for each independent force, so ##f_1=f_2=\ldots=f_n=f##.
With that, I think I've made a reasonable argument that the quantity in question ought to be ##\sum F_if(x_i)##, and with the additional assumption that ##f(0)=0##, that's what you started with.
Does that make sense?