($>) is analogous to (*>) not by dropping the f in the first position, but the f in the second, as proposed.
(*>) :: Applicative f => f a -> f b -> f b
($>) :: Functor f => f a -> b -> f b
It is unfortunate that (*>) is not equivalent to (flip (<*)), but it is a special case where breaking the "flipped characters" convention is useful for the sake of Applicative's convention to gather effects from left to right. It is nice that ($>) adheres to the flipped characters convention as well as being analogous to (*>).