Sorry, after a little more thinking I ended up un-accepting this answer (based on Piotr's comment). The issue is that a family of constant $\mathbb{N}$-motives over a variety $X$ may not be (e.g. étale) locally constant, since it may be twisted by invertible functions on the base. As an example, if $L$ is a line bundle on $X$ and $X_L$ is the normal log structure on $X$ induced on $L$, its Kato-Nakayama space is the circle bundle associated to $L$, which remembers at least the Chern class (in Betti homology): something that can be nontrivial if $X$ is e.g. simply connected.

Right, the log structure will be given by a local system of semigroups locally isomorphic to some power of the natural numbers. I don't think the local system will in general be trivial.

Thanks Leonid, this looks like exactly the context I was looking for!

I see -- is there a model category structure on simplicial pro-finite sets which takes care of the completion procedure?

More generally (following some ideas of Roman Bezrukavnikov and David Kazhdan) I want to study the category of Z_p-points of "semistacks" similar to $\text{pt}/\overline{G}_m$, whose representation theory should have nice properties.

The motivation is essentially #5. I don't know enough of this theory to check it, and don't know if the fppf local simplicial construction is "correct" (in particular, whether it gives the right answer for (5)) -- it would be great if it were!