Your question is why FATE can't time travel, right?
Well, then let me ask you a question in return: Why would it want to? Belthasar constructed FATE, there was no need for FATE itself to travel through time as Belthasar was already looking after everything.
Still, if FATE wanted to time travel, sure it could. It could just send Lynx (FATE's proxy, and actually FATE too) to get the Neo Epoch. However, I'm not sure if FATE will be able to control Lynx even in other time periods, where FATE might not exist. I don't think that would be possible. If you put it that way, that FATE loses control over Lynx when he travels through time, then it is impossible for FATE to travel through time.