The Last of Us was as much of a film as it was a game, and while that certainly irked some players, it did nothing to detract from the game being a masterpiece.
Part of the game's design was drafting the story and gameplay, and then once the game was on its way to being complete, combining that into a single script. The dialogue read like that in a film, and the gameplay was described in prose. This is covered in an interview with the game's director, Neil Druckmann, by the YouTuber Lessons from the Screenplay.
Set up from the very beginning to get the player invested in empathizing with Joel's story, Druckmann references the back and forth the team went on about how to make the beginning work. This started with having the player play as Sarah, Joel's daughter, and the developers succeeded in grounding the players in the game's setting of ferocious fungus zombies, making things a lot more intense.
What makes the game work most as a film is its use of story beats. Beats are moments in which something happens that connects points of a story, whether they be emotional or plot-based. In the case of The Last of Us, a great example of a story beat, involving Sarah, is towards the end of the intro mission. Joel has escaped the town, but runs into a soldier that fires on him, killing his daughter in the process. The beat is Sarah's death, which propels the story forward to the point where, years later, he meets Ellie.
Check out the video below.