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We study a generalized distributionally robust chance-constrained set covering problem (DRC) with a Wasserstein ambiguity set, where both decisions and uncertainty are binary-valued. We establish the NP-hardness of DRC and recast it as a two-stage stochastic program, which facilitates decomposition algorithms. Furthermore, we derive two families of valid inequalities. The first family targets the hypograph of a "shifted" submodular function, which is associated with each scenario of the two-stage reformulation. We show that the valid inequalities give a complete description of the convex hull of the hypograph. The second family mixes inequalities across multiple scenarios and gains further strength via lifting. Our numerical experiments demonstrate the out-of-sample performance of the DRC model and the effectiveness of our proposed reformulation and valid inequalities. [ABSTRACT FROM AUTHOR]