Hmmm, so even setting the destination target to “null” and then re-initializing the target position on the next frame doesn’t fix the odd NavMesh Agent behavior.
I’ve confirmed that the NavMesh has no holes or disconnects.
What’s further odd is that the line renderer confirms there are positions the agent is supposed to walk towards (even if the end point is wrong), but the agent is ignoring those, too.

In this image, the end point for the agent is south-west from the monster’s current position, but you can see the monster is heading directly west into the wall (represented by the z-axis arrow in blue).
Agents that are affected by the problem seem to have no obstacle avoidance, only 1 way-point in their navigation (i.e. no corners), and ignore even that 1 end way-point.
Enabling and disabling “Auto Repath” has no effect.
Enabling/disabling root motion on the Animator has no effect.
Different Update Modes on the animator has no effect, though “unscaled time” did teleport the monster far off the map.
Different Culling Modes on the animator has no effect.
Changing the obstacle avoidance quality on the agent has no effect.
Changing the area mask layer on the agent to “everything” has no effect.
There are no Off Mesh Links in my scene.
What FINALLY fixed the affected monster is disabling the NavMesh Agent component, and then re-enabling it.
I should specify that I am on Unity 2018.4.23f1.
Of course, I can code the agents to disable and re-enable themselves every so often, but I would much rather learn what is causing the problem to begin with so I can avoid it all together.
Anyone have any ideas?