Only one distance is ever truly in focus. Everything else forms a small disc rather than a point, and depth of field is the range where that disc is small enough to look sharp.
The size of that disc depends on the cone of light converging on the sensor. A wide aperture makes a fat cone, so a point at the wrong distance spreads into a large disc quickly. A narrow aperture makes a thin cone, and the same misplaced point spreads much more slowly.
So it is the angle of the cone doing the work, and the extra light is a side effect of the same opening rather than the cause.
This is also why a pinhole camera has effectively infinite depth of field, and why it needs such long exposures.