This Letter presents measurements of long-range transverse-momentum correlations using a new observable, v_{0}(p_{T}), serving as a probe of event-by-event radial-flow fluctuations, the underlying radial expansion, and the medium's properties in heavy-ion collisions. Results are reported for inclusive charged particles, pions, kaons, and protons across various centrality intervals in Pb-Pb collisions at sqrt[s_{NN}]=5.02 TeV, recorded by the ALICE detector. A pseudorapidity-gap technique, similar to that used in anisotropic-flow studies, is employed to suppress short-range correlations. At low p_{T}, a characteristic mass ordering consistent with hydrodynamic collective flow is observed. At higher p_{T} (>3 GeV/c), protons exhibit larger v_{0}(p_{T}) than pions and kaons, in agreement with expectations from quark-recombination models. Comparisons to viscous hydrodynamic calculations with varying bulk viscosity and equation of state demonstrate the sensitivity of the v_{0}(p_{T}) observable to these key medium properties. The findings establish v_{0}(p_{T}) as a valuable addition to the set of observables used in Bayesian analyses for extracting the transport properties and constraining the equation of state of strongly interacting matter, while also helping to systematically explore its sensitivity and impact within such global studies.