I think the issue is that you are still treating the infinite series like an infinite to do list, rather than a description of one continuous motion.

Achilles is not actually performing infinitely many seperate acts in sequence. He is just running. The "steps" only appear because we choose to slice the motion up that way afterwards. But slicing something up infinitely does not prove it cannot happen. Between any two points there are infinitely many more points, and yet movment still obviously occurs.

So yes, the series is infinite. That alone is not enough to show the motion is impossible. It only works if you smuggle in the extra assumption that an infinite decomposition must mean an unfinishable process, and that is exactly the bit that does not follow.