Measurement-free quantum error correction (MFQEC) offers an alternative to standard measurement-based QEC in platforms with an unconditional qubit reset gate. We revisit the question of fault tolerance for a measurement-free variant of the Steane code that leverages multi-qubit gates, finding previously overlooked phase-flip errors that undermine fault tolerance. We construct a revised MFQEC circuit design that is resistant to all single-qubit errors, but which nonetheless cannot tolerate certain correlated errors. In order to investigate fault tolerance systematically, we introduce an efficient method to classically simulate MFQEC circuits with (i) Clifford gates for syndrome extraction, (ii) ancilla-controlled Pauli operations for decoding, and (iii) a Pauli noise model. We thereby find a pseudothreshold of ∼0.7% for our revised MFQEC Steane code under a restricted noise model previously considered in the literature. We then relax noise model assumptions to identify general requirements for fault tolerance with multi-qubit gates, finding that existing multi-qubit neutral atom gates are incompatible with fault-tolerant syndrome extraction in both the measurement-based and measurement-free Steane code. We also find that decomposing multi-qubit gates down to a two-qubit gate set similarly spoils fault tolerance for the measurement-free Steane code. Finally, we discuss the theoretical ingredients that are necessary to recover fault tolerance for MFQEC codes.