[Duranage]

This tree stump east of the Aetheryte in East Shroud: http://i.imgur.com/uIN3ztw.jpg

It does not have a navmesh. As you can see in the picture, the boss (the giant insect on top of the stump in the picture) of a particular FATE spawns on it. When that happens, the bot just stands there staring at it forever.

Wouldn't matter even if it did. You can not make it onto the stump for it to require a mesh on it. Either add a ranged attack to your skill profile, or blacklist the fate and the monster.

Then something must be pretty odd, because I'm using Sebb's SMN profile, and I'm just standing there staring at the target.

I have never had an issue with that spot just using the basic SMN profile and running fates in the area. I would check with your skill manager profile as I have made tweaks to my PLD profile which used to do that fate.