  Quantum spin liquids (QSLs) are exotic phases of matter exhibiting long-range
entanglement and supporting emergent gauge fields. A vigorous search for
experimental realizations of these states has identified several materials with
properties hinting at QSL physics. A key issue in understanding these QSL
candidates is often the interplay of weak disorder of the crystal structure
with the spin liquid state. It has recently been pointed out that in at least
one important class of candidate QSLs - pyrochlore magnets based on non-Kramers
ions such as Pr$^{3+}$ or Tb$^{3+}$- structural disorder can actually promote a
$U(1)$ QSL ground state. Here we set this proposal on a quantitative footing by
analyzing the stability of the QSL state in the minimal model for these
systems: a random transverse field Ising model. We consider two kinds of
instability, which are relevant in different limits of the phase diagram:
condensation of spinons and confinement of the $U(1)$ gauge fields. Having
obtained stability bounds on the QSL state we apply our results directly to the
disordered candidate QSL Pr$_2$Zr$_2$O$_7$. We find that the available data for
currently studied samples of Pr$_2$Zr$_2$O$_7$ is most consistent with it a
ground state outside the spin liquid regime, in a paramagnetic phase with
quadrupole moments near saturation due to the influence of structural disorder.
