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This paper presents a search for generic short-duration gravitational-wave (GW) transients (or GW bursts) in the data from the third observing run of Advanced LIGO and Advanced Virgo. We use coherent WaveBurst (cWB) pipeline enhanced with a decision-tree classification algorithm for more efficient separation of GW signals from noise transients. The machine-learning (ML) algorithm is trained on a representative set of noise events and a set of simulated stochastic signals that are not correlated with any known signal model. This training procedure preserves the model-independent nature of the search. We demonstrate that the ML-enhanced cWB pipeline can detect GW signals at a larger distance than previous model-independent searches. The sensitivity improvements are achieved across the broad spectrum of simulated signals, with the goal of testing the robustness of this model-agnostic search. At a false-alarm rate of one event per century, the detectable signal amplitudes are reduced up to almost an order of magnitude, most notably for the single-cycle signal morphologies. This ML-enhanced pipeline also improves the detection efficiency of compact binary mergers in a wide range of masses, from stellar mass to intermediate-mass black holes, both with circular and elliptical orbits. After excluding previously detected compact binaries, no new gravitational-wave signals are observed for the two-fold Hanford-Livingston and the three-fold Hanford-Livingston-Virgo detector networks. With the improved sensitivity of the all-sky search, we obtain the most stringent constraints on the isotropic emission of gravitational-wave energy from short-duration burst sources.