Yup.
The point is that the Elementalist is a totally fine concept to play in 5E without over compensating him.
Would an Elementalist not take Detect Magic because it is not a fire spell?
Would an Elementalist not take Identify because it is not a fire spell?
Would an Elementalist not take Dispel Magic because it is not a fire spell?
Would an Elementalist not take Telekinesis because it is not a fire spell?
Would an Elementalist not take Web because it is not a fire spell?
Would an Elementalist not take Haste because it is not a fire spell?
If a player decides to only cast fire spells and to not cast other spells that his PC took (or to take a lot of spells that have a low utility and are rarely used), that's not an indication that the basic character concept is not fine. It might indicate that the given player is a bit of a loon.
The game design is just fine.