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We present measurements of the total and differential cross sections for near-threshold $J/\psi$photoproduction obtained with the CLAS12 detector at the Thomas Jefferson National Accelerator Facility. The results are based on data collected during the fall 2018 and spring 2019 running periods, using electron beams with energies of 10.6 and 10.2 GeV, respectively, scattered off a liquid-hydrogen target. Near-threshold $J/\psi$photoproduction offers a unique sensitivity to the strong interaction in the nonperturbative regime of quantum chromodynamics (QCD). The energy dependence of the cross section constrains the underlying $J/\psi$production mechanisms, including multigluon exchange and potential baryonic excitations. Additionally, the $t$dependence of the differential cross section can be related to the transverse spatial distribution of gluons in the proton, providing critical input for theoretical descriptions of the gluonic structure of the proton. An interpretation of the results in terms of the gluon content of the proton is presented, providing new experimental constraints on QCD-inspired models of the proton structure and the role of gluonic degrees of freedom in hadronic mass generation.