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We study the potential for precision electroweak (EW) measurements and beyond-the-Standard Model (BSM) searches using cross-section asymmetries in neutral-current (NC) deep inelastic scattering at the electron-ion collider (EIC). Our analysis uses a complete and realistic accounting of systematic errors from both theory and experiment and considers the potential of both proton and deuteron beams for a wide range of energies and luminosities. We also consider what can be learned from a possible future positron beam and a potential ten-fold luminosity upgrade of the EIC beyond its initial decade of running. We use the SM effective field theory (SMEFT) framework to parameterize BSM effects and focus on semi-leptonic four-fermion operators, whereas for our precision EW study, we determine how well the EIC can measure the weak mixing angle. New features of our study include the use of an up-to-date detector design of EIC Comprehensive Chromodynamics Experiment (ECCE) and accurate running conditions of the EIC, the simultaneous fitting of beam polarization uncertainties and Wilson coefficients to improve the sensitivity to SMEFT operators, and the inclusion of the weak mixing angle running in our fit template. We find that the EIC can probe BSM operators at scales competitive with and in many cases exceeding LHC Drell-Yan bounds while simultaneously not suffering from degeneracies between Wilson coefficients.