Well, 5e (and D&D in general) does encourage some types of specialization by dint of the class (and subclass) system. It just cannot accommodate all types of specialization, as indeed no system could. Specializing in a vary narrow type of magic--out side of say, school specialization--just does not work very well. Ideally, a special class or subclass could be made with class features and spell lists to make a balanced fire mage.
Or, lacking that, I think it could be a fair bit of re-flavoring to make things work. Even Gandalf the Grey, who specialized in fire & lights type of magic, still had other tricks to help out. Things like 'dispelling' hostile fire hazards or attacks "I am a Servant of the Secret Fire and a wielder of the Flame of Anor! The Dark Fire will not avail you, Flame of Udun!" Perhaps that Cone of Cold is merely a skilled fire mage taking the heat out of everything in the area. After all, Morgoth (to once again go to Tolkien), who was somewhat of a Fire God (or at least, that seemed to be his realm, just as Manwe was Air, Ulmo was Water & Alue was Earth) caused fierce colds to ravage the land and thereby created snow and ice. Perhaps a divination spell is seeing a vision in a fire burning special fuel with incense. Magic Missile and radiant damage can be seen as part of, or at least adjacent to, a broad defination of a fire theme.