I think the later reasoning about limits and infinitesimals proves Zeno right. After all, he certainly wouldn't have bet good money on the tortoise. In other words, Zeno is saying "there is a flaw in our understanding of how mathematics, distance, etc. work." His "problem" is not a paradox like "All generalizations are false, including this one." It's more of a koan, a challenge to future thinkers to refine our understanding of how reality (for lack of a better word) works. As you say, his problem questions our INTUITION of how things operate.