The mass of the earth dictates the acceleration of the individual masses towards the earth. However the acceleration of the earth itself towards the masses are dependent on how much mass is falling towards the earth. When more mass is falling to the earth, the earth accelerates towards the masses faster. So the thought experiment is flawed because with only one 1 kg weight falling towards earth, the gap between the weight closes slower than when there are three 1 kg weights spaced 1 m apart and dropped simultaneously.
If you define fall as the size of the gap. You could also take it as acceleration towards the barycenter, which would be the same. These are indistinguishable for everyday objects so could argue that the word “fall” could be interpreted either way.