If another cis man was suddenly uncomfortable with "they", I'd attribute that to some fragile sense of masculinity or a weird anti-woke crusade, as was the case with most such cases in my life, and find it personally annoying.
I don't want to deliberately make other people uncomfortable with my language, but it does feel exhausting to second-guess yourself, having to double-check if any innocuous phrase might be misinterpreted as an underhanded aggression, preempting a correction we know will be awkward for both sides.
I think we'd do better to consider context more than the specific language...