I think this encourages "look it up" and "this layout is just a convention" when one convention is much more "natural" i.e. at a point x, a dual element df(x) acts on vectors y via df(x)(y) = < gradf(x), y>.
I wouldn't teach it this way, but I would definitely take the taylor expansion and define the grad vector as the one to make the best local linear approximation. This tells you that the grad lives in the same space, i.e. same dimensions.
Of course, you can always switch things around if you want to calculate things and put in transposes where they belong. But I find it insane to take the unnatural view point as standard, which I find a lot of papers do.