Basically no, there is not. There is absolutely zero reason or purpose for it to fly, given the fact it is not actually the Fenrir, as in, the reoccurring summon monster, but.... just a plain, regular wolf that was trained to be used as a mount that was given the name of the breed's legendary alpha male (the end boss of Snowcloak).

Wolves both in real life and here in FFXIV do not fly, so neither does the mount. Besides, there are plenty of other flying mounts in the game to fly around with. Just because it costs 1 million MGP does entitle it to fly. That was just a time sink to stop players consuming the GS content rapidly simply to get the mount. But sadly like other attempts to force players to pace themselves it didn't quite work out that way.

Also, if you think there is no reason to visit the ARR zones... then you're only restricting your own enjoyment if all you do is sit in Idyllshire and spam endgame stuff for tomes. But that is a different topic to what's being said here so I'll say no more on my opinion on that.