Even in non-gaited horses, a horse that is in a "working walk" should have some amount of head-bob. That means its paying attention to you, not sleep-walking, dog-walking, or just trudging around in circles.
It's not as pronounced as what you'll see in Walkers, of course. . .but it's there.
Big Lick Walkers are a far cry from what the breed was intended for. I lucked-out when I bought my little Black Horse (in my avatar) - he's from BL lines but never made the cut for that show ring, and it turns out that he's a heck of an awesome trail horse. When I bought him, I knew very little about the difference between BL and naturally gaited walkers. But I would never consider buying another horse from BL bloodlines.
"Parelli horsemanship is just like painting by the numbers. You need absolutely no skill. You just put this color here and this color there, and when you're done, you have ... a mess no one wants." mp