Not a Storytelling Game

We Play to Discover
White Box, and by extension the later editions, is not a "storyteller" game at heart. Can you use it that way? Certainly, yes. There are lots of ways to play White Box, but as written, as designed, it is not a vehicle for the referee to plot out a story and have the players act through it using their characters. White Box is designed as a "wargame", just like it says on the cover. (the keyword is "game")
In the early 1970's there was a hobby called wargaming (it still exists in a somewhat diminished state). The hobby involved playing strategic games with maps and counters and sometimes with miniature figures and tabletop terrain. White Box is an extension of that idea in its original conception. Add some fantastic creatures, some magic spells and a character advancement system and that's the basis for the fantasy supplement to Chainmail and later the White Box. Stories are what the players tell after the game play is over.
The conversational nature of White Box quickly led to many players wanting to play the role of their character as we identified with "our guy". In response to the referee asking "What do you want to do now?" some responded that their character wanted to talk to someone/ something. The player stated what the character would say and the referee responded with what the NPC/ monster replied. And so "role-playing" takes on a more nuanced aspect and rather than just moving and fighting as figures had done in wargames, we began having "in character" conversations.
White Box is not a storyteller game because players have agency. (I know that may sound a bit odd, but stay with me.) Each player controls the actions of their own character, therefore they do not follow a script as actors in a play. The players choose how their characters will react to the referee's milieu and world. The referee describes a situation, the surroundings and actions of any non-player characters. The players then determine how the main actors, their characters, will behave. No one really has control of the ensuing story. It just develops as a result of the character actions and random outcomes of the dice.
White Box is designed to uncover lots of stories with different themes, involving different worlds and different personalities, some heroic, some not. The rules imply a certain type of fantastic setting with alignments and magic, both arcane and divine, often involving fantasy beings taken from myth and literature who have adventures involving wilderness and underground places of mystery. It does not include mechanics to facilitate storytelling. There are no devices to allow players to interject new fiction into the setting. Players control the actions of their characters in a setting controlled by the referee. It is that way as part of its roots in wargaming where the map or table terrain, the troops, the objectives and victory conditions are all static. Can we add elements to White Box allowing for a more storytelling style of play? The answer is of course, "Yes, certainly", but in doing so it changes the nature of the White Box game (and perhaps makes the game more to your liking by doing so).
We play White Box to find out what will happen? Discovery, conflict and adventure... and sometimes glory are the result. White Box is a creative outlet for our imagination and a temporary escape from the everyday world. Taken to an extreme it can be bombastic epic, humorous foolishness or grim and perilous nihilism. A strength of the original design is how malleable the White Box rules are. They easily allow for customization and the exploration of lots of different play styles. I would say they encourage it.