Mechanics are vital in that they establish what a setting is like, and how the inhabitants fit into the setting. This is true whether the setting is explicit, or implicit. That is, whether the setting is detailed and presented as such to the participants, or implied in the structure and functioning of the rules. Without mechanics, without a system a game has no form, provides no guide as to expectations regarding what behavior is acceptable and unacceptable in the context of the game. And with no guide as to how characters are expected to behave players are lost and find it hard to give direction to their character's lives.
It's not just a matter of our needing rules, we live by a set of implied rules we follow. Even when we think we are acting purely at whim. At the core we live by instincts we were born with that govern our behavior around others, even when we are certain we are acting of our own free will. This is even more true of our entertainments, where instinct can conflict with what our culture teaches.
Without mechanics the player has no reliable way to learn how he, as his PC, is expected to behave in the setting. To learn what is allowed and what is expected of him. Mechanics establish what is and is not possible, what is and is not allowed. Without mechanics a game becomes naught more than chaos, and chaos taken to its ultimate expression is nothing more than a dull, featureless waste.