Exactly why it would be a pleasant surprise to actually find one. There's an iguana involved in the nomad origin story. If I recall right those are considered extinct too but here it is - a living iguana. OK, so let's say dogs are off the table for some reason. Why not let us keep the iguana?
Apart from the obvious reduction in development time needed, why stick to a single animal species for the entire game?
There's this shard that tells us about three Japanase words: kiji (pheasant), hato (dove) and taka (hawk) and relates them to members of the Arasaka family as a metaphor for their attitude towards tradition. Why not incorporate at least one of those into, let's say, the corpo origin story?
I don't know. Maybe I'm asking too much...