  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 $\langle
\theta_{\rm j} \rangle \approx 6.1^{\circ}$ [-3.2$^{\circ}$,+9.3$^{\circ}$]
(68\% confidence on the full distribution) from jet measurements alone. For the
remaining events, we infer $\theta_{\rm j}\gtrsim 0.5-26^{\circ}$. We uncover a
population of SGRBs with wider jets of $\theta_{\rm j} \gtrsim 10^{\circ}$
(including two measurements of $\theta_{\rm j} \gtrsim 15^{\circ}$),
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.
