I personally think that it's a mistake to interpret gravity as being identical to spacetime, and thus mass as deforming spacetime just as the analogous case where a bowling ball bends a rubber sheet. I think then we're basically talking about container space (a legacy of Newtonian physics) and one thing that general relativity shows, I believe, is that spacetime does not contain massive objects.
Remember that the gravitational field couples universally since it is generated, and effects, massive objects. Every object we can empirically experience has mass (mass-energy) so every object generates, and is influenced by, a gravitational force. Because of this, we can then use the gravitational field to define relative distances and relative acceleration. These relative distances and relative accelerations come in the form of the metric.
So gravity is not identical with spacetime. Gravity constitutes spacetime. What we perceive to be spacetime is a phenomenological manifestation of the gravitational field.
So why do we "perceive" spacetime to be warped? Because that is the way the gravitational field propagates.
