Achilles and the tortoise paradox?
If you start 1 meter ahead of me, and it takes me 1s to reach your current position (apparently I run at 1 meter per second, and you run at .01 meters per second), $\frac{1}{100}$th of a second to reach your position at $t=1$, etc. I take $$1 + \frac{1}{100}+\frac{1}{100^2} + \cdots = \sum_{i=0}^{\infty}\frac{1}{100^i}$$ seconds to overtake you. Since $$\sum_{i=0}^{\infty}\frac{1}{100^i} = \frac{1}{1-\frac{1}{100}} = \frac{100}{99},$$ then after $\frac{100}{99}$ seconds, I will have overtaken you. This will occur well before we reach the finish line; we've only advanced $\frac{100}{99}$ meters (since apparently I run at 1 meter per second), and the finish line is more than $\frac{100}{99}$ meters from where we started. After I've overtaken you, I will be ahead at any further time.
The implicit error in the original claim that I cannot overtake you is the assumption that an infinite sum of positive quantities will necessarily be infinite. This has long been dealt with, and does not even require the use of infinitesimals.
Of course, it's possible for you to start so far ahead of me that I will only catch up to you when we get to the finish line; but that's hardly a paradox! Nor do I understand what your complaint is with "both people finish the race". Is there some problem with the slower person finishing after the faster one has?
Below are some pointers to the literature for your philosophical concerns, taken from old sci.math posts of mine. Thus, this is really in the nature of a comment, not an answer, but because of comment length constraints I'm posting this as an answer.
First, of possible interest are the google searches infinity machines and supertasks, as well as the Wikipedia article Supertask.
I've listed the references that follow in order of how helpful/interesting I think they'd be for your concerns.
[1] Wesley Charles Salmon (editor), Zeno's Paradoxes, Bobbs-Merrill, 1970, x + 309 pages. [Reprinted by Hackett Publishing Company in 2001; ?? + 320 pages.]
[2] José Amado Benardete, Infinity. An Essay in Metaphysics, Clarendon Press, 1964, x + 289 pages. scanned copy
[3] Adolf Grünbaum, Modern Science and Zeno's Paradoxes, Wesleyan University Press, 1967, x + 148 pages. [Reprinted by George Allen and Unwin in 1968; x + 153 pages.]
[4] Florian Cajori, The history of Zeno's arguments on motion (in 9 parts), American Mathematical Monthly 22 (1915), 1-6, 39-47, 77-82, 109-115, 143-149, 179-186, 215-220, 253-258, 292-297. Jstor AMM Volume 22 issues: 1 2 3 4 5 6 7 8 9
[5] Florian Cajori, The purpose of Zeno's arguments on motion, Isis 3 #1 (January 1920), 7-20. jstor
[6] Clive William Kilmister, Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number, Mathematical Gazette 64 #429 (October 1980), 149-158. jstor
I think the mathematical "explanation" of the Zeno's paradox (convergence of infinite series) is quite unsatisfying. Assuming that each term in the series corresponds to one step of Achilles's and considering that he indeed overtakes the turtle in finite time, which of Achilles feet is forward at the moment when he reaches the turtle?
Or a slightly different, but equivalent, presentation of the paradox: assume that the turtle changes direction at each discrete instant of time Achilles reaches her previous position, alternatively moving NE and SE. Achilles just follows her path. What direction is the turtle facing the moment Achilles reaches her?
Achilles's and the turtle is no paradox at all, but a refutation of the hypotheses that the space is continuous. Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. Together they form a paradox and an explanation is probably not easy. For Zeno the explanation was that what we perceive as motion is an illusion. In any case, I don't think that convergent infinite series have anything to do with the heart of Zeno's paradoxes.
EDIT: The same argument can be made point-like particles, only assuming that physical reality is continuous and infinitely divisible. Imagine a photon travelling between an infinite sequence of mirrors placed in a zig-zag shape with distance between mirrors decreasing at a geometric rate. So the photon bounces from a NE to SE direction and back, with the distance travelled decreasing "fast". Since the length of the total path is finite (sum of a geometric series), the photon will emerge from the sequence of mirrors in finite time. What direction will it travel? The heart of the Zeno's argument is that there is no logical way to decide that. You may argue that it is impossible to build such a sequence of mirrors, however this is just conceding Zeno's point that physical reality is NOT continuous and infinitely divisible.
I think the mathematical model of the Zeno's paradox is a great pedagogical tool in first year calculus, probably could be made even earlier in high school, but it misses an important aspect of Zeno's argument. Granted, this argument lies at the boundary of math, physics and perhaps philosophy.