This can't be the sole answer to the problem of gravity, because a model like this would make modeling the solar system difficult -- if you eliminate the gravitational force from ordinary matter and replace it with a repulsive force from dark matter, where do you put the dark matter so that every planet and asteroid in the solar system obeys a $GM/r^2$ force law? We know that the answer can't be "in a shell at some great radius" due to Gauss's law, which also tells us that the answer can't be "distributed uniformly through the solar system", because this would give a different effective mass for Jupiter than it would for the Earth.
So, you still need ordinary gravity to explain the solar system. So, then we'd be using this repulsive dark matter to explain the stuff that ordinary dark matter does, but why would we posit an exotic gravitational force for something that we can already explain with ordinary dark matter obeying an ordinary gravitational force?