It's not so much that it's *necessary* as that it's *possible*. The existence of two functions in Data.Traversable explains both of the superclasses of Traversable:

fmapDefault :: Traversable t => (a -> b) -> t a -> t b

foldMapDefault :: (Traversable t, Monoid m) => (a -> m) -> t a -> m

Each of these is written using only traverse, and they can be used to define fmap and foldMap for types when you've written traverse.

Hint: Consider traversing using the following applicative functors:

newtype Const a b = Const a
instance Monoid a => Applicative (Const a)

newtype Identity a = Identity a
instance Applicative Identity