Field-free superconducting diode effects require both inversion-symmetry breaking and an internal time-reversal-symmetry (TRS) breaking field, making them sensitive probes of hidden order in superconductors. In centrosymmetric kagome AV₃Sb₅, the inversion symmetry generally should generally preclude the observation of the superconducting diode effect. Furthermore, though TRS breaking has been reported in the superconducting regime of CsV₃Sb₅, whether it is generated by superconductivity or inherited from charge-density-wave (CDW) order remains unresolved. Here we show that pristine CsV₃Sb₅ devices exhibit no intrinsic field-free superconducting diode effect, whereas surface oxidation or asymmetric etching activates a large nonreciprocal supercurrent. Moreover, the response is stochastic, with sweep-dependent polarity and magnitude, indicating metastable TRS-breaking domain configurations. Small out-of-plane magnetic fields stabilize the superconducting diode response, consistent with field selection of such domains. Finally, when long-range CDW order is suppressed by Ti doping, the SDE disappears. Our results establish the symmetry requirements for the field-free SDE in CsV₃Sb₅, reveal its stochastic domain-controlled character, and link superconducting-state TRS breaking to CDW-related order.