So while I was waiting for the fate to spawn, I did some looking around. A bunch of people in the comments of XIVDB were saying how you need to kill the nearest dragonfly to Stone Vigil for the fate to spawn. I went and did that and the fate spawned almost instantly. Thanks to this I ended up waiting no more than 10 minutes XD

This could be a coincidence seeing as many of you guys have reported cleaning the mobs and it still not spawning, but it worked for me so I figured I'd share.