It is mainly air. Headcuts in gullies are relatively easy to fix in their early stages, but can quickly grow into insatiable soil eating monsters that are costly to repair. There is also the claim that trees compete with grasses for groundwater. Tree roots pull water from deep in the ground. We're lucky if our grass roots extend eight inches deep.
Many trees are still dormant in the early growing season leaving water available for grass.
While it is true that surface moisture moves down to replenish deep soil moisture, annual grasses mature before this effect is perceptible. I have also noticed that some of our lushest pasture is underneath trees where the shade retains morning dew and prevents continuous sun from drying the soil. Trees also serve as windbreaks which lessen the drying effect of the wind. Periodic flooding of lowlands deposits sediment and minerals that increase the fertility of pastures. One can take advantage of this natural fertilization process by leaving enough residual dry matter or allowing sufficient new growth to slow down flood flows.