Why is "Answers" not allowed in the FC Recruitment vid contest?

[Enkidoh]

My guess, and this is a total shot in the dark, is that Answers was composed by Nobuo Uematsu whereas all of the other music in FFXIV is Soken.

This is not actually correct - a significant portion of FFXIV ARR's OST was composed by Uematsu, specifically tracks that returned from the original version of FFXIV (version 1.0), which ironically includes Answers. Although like you I have no idea either as to the reason why exactly SE has forbidden Answers to be used, but I'm going to make a wild guess that it probably has something to do with record licensing of the track.

Even though SE owns the copyright on it, they're probably wanting to keep it reserved for potential licensed use elsewhere by authorized third parties - hence free use by players in a competition would naturally conflict with this. But that's a real long shot on my part.