We measured the cross section of coherent elastic neutrino-nucleus scattering (CEvNS) using a CsI[Na] scintillating crystal in a high flux of neutrinos produced at the Spallation Neutron Source at Oak Ridge National Laboratory. New data collected before detector decommissioning have more than doubled the dataset since the first observation of CEvNS, achieved with this detector. Systematic uncertainties have also been reduced with an updated quenching model, allowing for improved precision. With these analysis improvements, the COHERENT Collaboration determined the cross section to be (165+30−25)×10−40 cm2, consistent with the standard model, giving the most precise measurement of CEvNS yet. The timing structure of the neutrino beam has been exploited to compare the CEvNS cross section from scattering of different neutrino flavors. This result places leading constraints on neutrino nonstandard interactions while testing lepton flavor universality and measures the weak mixing angle as sin2θW=0.220+0.028−0.026 at Q2≈(50 MeV)2.