Of course, airfields all started off as grass fields.
As aircraft got bigger & heavier a hard surface was needed as a runway. These hard runways were constructed on the grass airfields, the remainder of which remained as grass.
I guess that no one ever felt the need to pave over the rest of the airfield. To do so would have led to a blurred distinction between the runway & the rest of the field. Also, to have paved the whole area over would have been extremely expensive, & totally unnecessary.