Without going into your main question, note that photons do not necessarily apply only a pushing force; they also are responsible for the attractive, as well as repulsive, forces of magnets and charged particles. Why (and whether) gravitational forces are always attractive is a more intriguing matter.

Whether gravitational forces must logically manifest themselves in the form of particles is not as far as I know a settled question. If gravitational forces do indeed  amount to warping of space, then their effects might well  propagate at the speed of light if the nature of space is a function, consequence, or cause of the speed of light. The effects and causes would after all, not be independent, so it could amount to the same thing.  

Concerning the problem with gravitons, in 1992 Roy J. Britten wrote (Proc. Natl. Acad. Sci. USA, Vol 89, pp. 4086): "The primary unexplained assumption in this model is that after emission from one mass a graviton...is absorbed by a second mass [and] the momentum transfer is in the correct direction to cause an attractive force on the absorbing mass towards the emitting mass. The underlying mechanism is not easy to visualize, but the problem arises for any model that includes gravitons." It is actually not difficult to "visualize." But it doesn't make any sense. Nobody has ever given a physical explanation as to how the resulting momentum could be in the opposite direction of the initial momentum. Gravitons are an abstract human invention. Therefore, the question is like asking how Santa Claus flies around the world in a sleigh pulled by reindeer.