Why can't you assume time symmetric billiard balls? If you take a lot of them, black and white, put all the black ones on one side and the white on the other, give them random velocities, and wait a bit, you will end up with them all mixed together. The entropy has increased even though all interactions are reversible. The macroscopic transformation is time asymmetric (the balls will not sort themselves back again) even though the microscopic transformations are time symmetric (elastic collisions and frictionless movement).

The way I think about this (from Feynman) is that entropy measures "how special" a macroscopic state is, on average. When you apply many microscopic independent fluctuations to a given state you are unlikely to end up with a "more special" state. That's how states are ordered at the macroscopic level giving the arrow of time a clear direction.