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We present a comprehensive study of 29 short gamma-ray bursts (SGRBs) observed $\approx 0.8-60$ days post-burst using $Chandra$ and $XMM-Newton$. We provide the inferred distributions of SGRB jet opening angles and true event rates to compare against neutron star merger rates. We perform uniform analysis and modeling of their afterglows, obtaining 10 opening angle measurements and 19 lower limits. We report on two new opening angle measurements (SGRBs 050724A and 200411A) and eight updated values, obtaining a median value of $\langle θ_{\rm j} \rangle \approx 6.1^{\circ}$ [-3.2$^{\circ}$,+9.3$^{\circ}$] (68\% confidence on the full distribution) from jet measurements alone. For the remaining events, we infer $θ_{\rm j}\gtrsim 0.5-26^{\circ}$. We uncover a population of SGRBs with wider jets of $θ_{\rm j} \gtrsim 10^{\circ}$ (including two measurements of $θ_{\rm j} \gtrsim 15^{\circ}$), representing $\sim 28\%$ of our sample. Coupled with multi-wavelength afterglow information, we derive a total true energy of $\langle E_{\rm true, tot} \rangle \approx 10^{49}-10^{50}$\,erg which is consistent with MHD jet launching mechanisms. Furthermore, we determine a range for the beaming-corrected event rate of $\mathfrak{R}_{\rm true} \approx360-1800$ Gpc$^{-3}$ yr$^{-1}$, set by the inclusion of a population of wide jets on the low end, and the jet measurements alone on the high end. From a comparison with the latest merger rates, our results are consistent with the majority of SGRBs originating from binary neutron star mergers. However, our inferred rates are well above the latest neutron star-black hole merger rates, consistent with at most a small fraction of SGRBs originating from such mergers.