Yes.
> Don't most websites built with Foundation's grid system automatically fit nicely on mobile devices?
Yes, but responsive designs always means tradeoffs, and those tradeoffs will be different depending whether you're "desktop first" and scaling down or "mobile first" and scaling up. And of course, the framework can set a course on what to check and where to start from.
Luke Wroblewski is a big advocate of tackling responsive design from the bottom ("Mobile First"): http://www.lukew.com/ff/entry.asp?933 http://www.abookapart.com/products/mobile-first and he apparently pushed for Foundation 4 to much better support this design style.
I think it's a matter of orientation. Rather than creating a Desktop site and then scaling it down, you're creating a Mobile site and scaling it up.
I haven't really put much thought into it, but this may actually make more sense, since you're creating the minimum viable product first, and then adding to it, rather than creating a full project and then trying to figure out what to hack away.
Doing mobile first forces you to distil your app to just exactly the right components you need to achieve your product goal because there's so little real estate. It prevents you from adding non essential bullshit that you could easily throw into your design if you started with web.