But the video game analogy fails, because the real world is not made of pixels- it is continuous. That's the whole point of the dichotomy- for any given interval, you can always halve the interval. There is no atomic unit at which point further division is impossible (at least, conceptually impossible- certainly there are intervals that are not practically or physically divisible).
So, the dichotomy doesn't require the assumption that space is infinite (i.e. infinitely large), only that it is continuous and therefore infinitely divisible. And what, exactly, is paradoxical here depends on the construction of Zeno's argument- on the classic Aristotelian interpretation, we end up with an infinite number of sub-intervals, each taking some positive (non-zero) duration to traverse. If we have to do an infinite number of tasks, each with some non-zero duration, then it would take us an infinite duration- we would never arrive. Or so the argument goes. Obviously, contemporary maths to the rescue here. But one other construction of the paradox is to conclude that there is no FIRST interval, such that the traversal of which sees us begin our journey, and no LAST interval, the traversal of which sees us arrive at our destination. Though this is more of a counter-intuitive result rather than any contradiction, it can't be dealt with quite as easily as the traditional version of the paradox.