Hydride molecules lie at the base of interstellar chemistry, but the synthesis of sulfuretted hydrides is poorly understood and their abundances often crudely constrained. Motivated by new observations of the Orion Bar photodissociation region (PDR) – 1″ resolution ALMA images of SH⁺; IRAM 30 m detections of bright H S, H S, and H S lines; H₃S⁺ (upper limits); and SOFIA/GREAT observations of SH (upper limits) – we perform a systematic study of the chemistry of sulfur-bearing hydrides. We self-consistently determine their column densities using coupled excitation, radiative transfer as well as chemical formation and destruction models. We revise some of the key gas-phase reactions that lead to their chemical synthesis. This includes ab initio quantum calculations of the vibrational-state-dependent reactions SH⁺ + H₂(v) ⇄ H₂S⁺ + H and S + H₂ (v) ⇄ SH + H. We find that reactions of UV-pumped H₂(v ≥ 2) molecules with S⁺ ions explain the presence of SH⁺ in a high thermal-pressure gas component, P_th∕k ≈ 10⁸ cm⁻³ K, close to the H₂ dissociation front (at A_V < 2 mag). These PDR layers are characterized by no or very little depletion of elemental sulfur from the gas. However, subsequent hydrogen abstraction reactions of SH⁺, H₂S⁺, and S atoms with vibrationally excited H₂, fail to form enough H₂S⁺, H₃S⁺, and SH to ultimately explain the observed H₂S column density (~2.5 × 10¹⁴ cm⁻², with an ortho-to-para ratio of 2.9 ± 0.3; consistent with the high-temperature statistical value). To overcome these bottlenecks, we build PDR models that include a simple network of grain surface reactions leading to the formation of solid H₂S (s-H₂S). The higher adsorption binding energies of S and SH suggested by recent studies imply that S atoms adsorb on grains (and form s-H₂S) at warmer dust temperatures (T_d < 50 K) and closer to the UV-illuminated edges of molecular clouds. We show that everywhere s-H₂S mantles form(ed), gas-phase H₂S emission lines will be detectable. Photodesorption and, to a lesser extent, chemical desorption, produce roughly the same H₂S column density (a few 10¹⁴ cm⁻²) and abundance peak (a few 10⁻⁸) nearly independently of n_H and G₀. This agrees with the observed H₂S column density in the Orion Bar as well as at the edges of dark clouds without invoking substantial depletion of elemental sulfur abundances.