Strategic Planning, anyone?

What you describe as "strategic consulting" sounds a lot like the management consulting that firms like McKinsey, Bain, BCG, etc. do. You would get to do the things you describe, but would naturally be working like a consultant.

On the internal side, I think you'll find few companies that have a dedicated "strategic planning" group, and if they do it's usually comprised of pretty senior people in the organization doing navel gazing and thinking big thoughts. Most organizations I've experienced have a more fluid approach, with teams being pulled together from different parts of the organization to address specific projects initiated by executives. You'll also find bits of strategy in different departments, so if you're on a brand team you'd be working on strategy for a specific product, if you are in operations you may work on supply chain strategy, etc. I think the best way to get at some of that strategy work early in your career while getting some experience in different areas (other than doing consulting), is through a general management rotational program, or if you know what functional area interests you most a rotational program focused on that area.