In the “paternal exercise” case the trait isn’t the habit of exercising, it’s the metabolic changes of exercise that are (apparently) conferred to both father and offspring.

You're completely missing the point that I explicitly stated. The trait that is purportedly being transmitted is metabolic changes that confer some advantage, but that's not what's being measured in the father, "exercise" is--not a word was said about fathers having better metabolic health, just that they "exercise". Which is reason to be skeptical of the claim.

Fair enough. It would be a clearer result (more “apples to apples”) if the same trait was directly measured “on both sides”. Thanks for restating.

Edit: typo

Have you considered that checking for that specifically might not be actually needed? After all, the correlation between exercise and metabolic health is well established.

That was my original thought, however if you want to quantify an effect it would be ideal to measure the same trait in both parent and offspring. I assume that the reason that this was done (I didn’t read the papers) is because this is a retrospective study, where the participants self-report on the exercise level of their fathers, rather than a longitudinal study which could measure the fathers’ metabolic state fitness at time of conception.

Thus, although there is a plausible link, I now agree with the parent post that this is sufficient reason to take the study with a grain of salt (given well known academic incentives to produce positive results, etc).