I found this interesting problem and was able to find the solution by searching Elysiun. But it sure ain’t obvious, so for the benefit of the next guy who’s whacking his head against this problem (yeah, you! Hi! I’m talking to you!…) here’s the problem and the answer.
The problem is that my character wouldn’t actually walk when I specified “Use Path” in the NLA editor entry for the action. He’d slide across the path but wouldn’t walk. I also observed that if the “Use Path” button was de-selected, he’d walk, but of course wouldn’t follow the path.
The not-so-obvious “solution” is that, when you parent the armature to the path that the armature is to follow, you must not specify the obvious, “Follow Path.” You must, instead, use the other option … normal parenting.
Now, personally, I think this is a “bug” in the sense that the NLA action shouldn’t care how the affected object has been parented to the path. At the very least it should not respond by inexplicable behavior. When an action specifies that a character should walk, along a path, and it has been (by any plausible means) parented to a path, it should … walk that path.
But it doesn’t, and here’s how to fix it, and “hope this helps.”