This is probably one of the most-least-often-asked questions, so good job asking the tough questions. Most people just take things for granted.
Being mindful of time or the illusion thereof is necessary for managing one's energies, as one usually has multiple things on their short and long term to-do lists. In other words, we always have something else to do later, so we need to set a time limit on the game so that we are not infinitely giving our sum total existence to a single game or a single move in a game.
Of course, there's the concept of relative simultaneity which says that there really isn't any such thing as time, just everything happening all at once, in relation to everything else happening all at once. It's called being. The universe is being, and you are part of the universe. We all are.
Why do we need time in chess