In ordinary English we sometimes assign gender to a behavior and not sex, e.g. in "x throws like a girl," "x" need not necessarily be a girl, the particular throwing motion is what gets the gender. And we also speak of "acting like a lady" and "behaving like a gentleman." Some people, e.g. Mutt, speak of "men's work" and "women's work." In our everyday usages, right and left, we assign gender when not referring to biological sex. This is more striking in languages that assign genders to all nouns, even new words.
The problem is "acting like a lady" is subject to interpretation and the lack of a universal criteria. The way I see trans- people is that they identify with what they understand to be the opposite sex's "role."
If someone would like to introduce a strict definition where the concepts are synonymous, that is fine, but it is not in line with ordinary usage and I believe this distinction is what is being appealed to. I believe the distinctions between the usages of 'gender' and 'sex' are empirical, but whether the distinction should exist or not really comes down to your ethical disposition. Most people appeal to a 'moral' or 'natural' or 'supernatural' world order to justify their disposition.
Now it would be absurd to us to hear someone say pregnancy can be a "man's job," say if a trans-male got pregnant, but the understanding of biological facts doesn't seem to me to be what is in contention, but rather the new and unique (to us) usage of the word 'Man.' And the "can" in "can be a man's job" refers to a 'logical possibility,' not a physiological possibility.