It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead.
Only the parts of the story that support the paradox are adequately stated. It doesn't say that Achilles has to stop at each point, which is left as an implication(?). If he doesn't than there is no paradox.
The reaction time of Achilles each time he stops isn't specified, so:
Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.
If Achilles does stop at each point, he is in essence forever chosing to allow the turtle to stay ahead, which is no paradox either.
The other implication is that Achilles *must* go only 1/2 the distance, and no further, but we'll let that slide, since that's the good part.
Zeno could simply have said, if you halve something and halve the result, and repeat, would you ever have nothing left? This is the clear idea behind it all, but instead Zeno chooses a clever wording that creates an apparent paradox out of a straightforward concept, the paradox being "Achilles can never reach the turtle". There is no paradox in the previous statement of infinite reduction, only the question of whether you think there is an indivisible limit or not.
So, Zeno's "proof" only served to cloud and confuse, which was what he was attempting, since he was in a charged debate, not actually trying to find the answer.
On the other hand,
In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...
— Bertrand Russell, The Principles of Mathematics (1903)1
So, if you're still in awe of the paradoxes, you can count him on your side.
Personally, I think he's just being contrarian.
Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite.
So much for being the "foundation of a mathematical renaissance" after two thousand years. Not only that, but it was the "failure" of his proofs which stood the test of time, so giving him the credit for the success in the area of infinities is accidental at best. Remember he was trying to show the falacy of infinitely divisible space in reality.
The paradoxes lasted this long because they were cute, and had real world objects in them doing odd things. Zeno was just an early spin doctor, a master of sound bites. He didn't invent the concept, just stuck words around it. People thought they were fun, so they were retold, and of course they mystified the layman. Aesop's fables lasted for the same reason, but they were a lot better.
Blah, the more I think about it, the more I dislike him. The world is full of disingenuous people doing things like that, using their brilliance to warp ideas to suit their purposes, and is very much the worse for it.