Exactly. Genetics does play a role but some website that alludes to 'nope nothing you can do if you or your spouse have allergies because the kid has a genetic predisposition to having allergies in the first place' (which cites zero studies) and leaving it at that....gives folks the impression that it has nothing to do with environment or behavior which is patently false. Tells less than half the story. As websites like that do.
Story you linked gets it exactly correct in terms of making sure the whole story is presented.
Edit: this all of course doesn't even BEGIN to discuss the fact that 'genetic predisposition' in and of itself is hilariously misleading because most research seems to indicate that if there is a genetic link between environmental allergy predispositions that its likely controlled by MULTIPLE genes all of which, (once identified) would then have to be examined for dominance/recession/co-dominance/epistasis etc in terms of how they interact with each other, the result of which would be phenotypically distinct in order to create 'levels of predisposition' if such a thing actually existed. But I'm sure baby center's got that covered somewhere.