Is it silly to level an Alt?

[Niwashi]

Is there enough content to level all classes on the same character?

There isn't enough quest content, because regular quests can only be done once per character, so the ones you've used on one class are not available for leveling other classes. However, there are several types of repeatable content available: guildleves, guildhests, dungeons, and (once you're into the upper levels of the game) beast tribe quests. Plus, each class gets its own class quests and hunting log. The combination of all of them provide plenty of content for as many classes as you want to level.