Why does (NH4)4SO4 + AgNO3 make no reaction?

Perhaps you did something wrong - I cannot be sure. However, I would like to point out that Ag2SO4 is slightly soluble in water, to the extent of 1.2 grams per 100 mL of water -- by far more soluble than AgCl. Therefore, if the concentrations of Ag+ and [SO4]2- were sufficiently small in the mixed solution, no precipitate should have formed.