Contrary to the widespread belief that Majorana zero-energy modes, existing as bound edge states in two-dimensional topological insulator (TI) -superconductor (SC) hybrid structures, are unaffected by nonmagnetic static disorder by virtue of time-reversal symmetry, we show that such a protection against disorder does not exist in realistic multichannel TI/SC/ferromagnetic insulator (FI) sandwich structures of experimental relevance since the time-reversal symmetry is explicitly broken locally at the SC/FI interface where the end Majorana mode (MM) resides. We find that although the MM itself and the bulk topological superconducting phase inside the TI are indeed universally protected against disorder, extra fermionic subgap states are generically introduced at the TI edge due to the presence of the FI/SC interface as long as multiple edge channels are occupied. We discuss the important implications of our finding for the detection and manipulation of the edge MM in TI/SC/FI experimental systems of current interest.