If you follow your Wikipedia link and then follow the link for Green's function two times, you come to the article that points out that Green's function methods only work on linear differential equations. The multipole method you refer to is based on the Green's function and only works for linear differential equations. Einstein's equations are non-linear. Trying to quantize a non-linear theory leads to problems that no one has yet resolved. You should also note that non-linear differntial equations do not support superposition; that is, the solutions cannot be added to yield another valid solution. They are not additive. So, the method you suggest fails because it does not meet the criteria you refer to.

Gravitational interactions become stronger with decreasing distance.

Take the Newtonian gravity model, if force is proportional to 1/distance^2, as distance approaches zero, the force approaches infinity.

You can simplify all points in a region as being one point for points that are far away, but that doesn't help with the infinite force problem with close points.