YES fates are broken.

[StarHate]

No one told you you had to do Animus. You knew what the requirements were and what might happen (including rare FATES.) If that wasn't something you were on board with why sign up? There are plenty of other weapon alternatives for those that don't have the time or the will to fulfill the relic weapon requirements.

You seem to be missing the entire point. In the last couple patches they changed it so that the longer it has been since a fate spawned, the more likely it is to spawn. If you are clearing the area, no single fate should take longer than 3 hours to spawn. The OP is saying that isn't happening as she waited for more than 3 hours clearing fates and it didn't pop. Wanting or not wanting to do her animus has 0 correlation with the change they made to fates not working.

edit - I went and checked, it is the 2.3 patch notes: "The chance of a FATE occurring will increase the more time passes since its last occurrence, increasing the overall frequency of FATEs."