The stability of the Dirac spin liquid on two-dimensional lattices has long been debated. It was recently demonstrated [Nat. Commun. 10, 4254 (2019)10.1038/s41467-019-11727-3 and Phys. Rev. B 93, 144411 (2016)PRBHB72469-995010.1103/PhysRevB.93.144411] that the staggered π-flux Dirac spin-liquid phase on the nonbipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase is experimentally viable. For SU(2) spin systems, the effective zero-temperature low-energy description of Dirac spin liquids in (2+1) dimensions is given by the compact quantum electrodynamics (cQED2+1) which admits monopoles. It is already known that generic quenched random perturbations to the noncompact version of QED2+1 (where monopoles are absent) lead to strong-coupling instabilities. In this paper we study cQED2+1 in the presence of a class of time-reversal invariant quenched disorder perturbations. We show that in this model, random non-Abelian vector potentials make the symmetry-allowed monopole operators more relevant. The disorder-induced underscreening of monopoles, thus, generically makes the gapless spin-liquid phase fragile.