So, Zeno assumes that, to go from the mark at $1m$ to the mark at $2m$ we've to do an infinite number of tasks. Like the task of getting to $1.001m$, the task of getting to $1.000005m$,the task of getting to $1.658m$,etc.
So, to perform an infinite number of tasks, one must take an infinite amount of time. I think that's where the loophole is. Well, yes, that would be true only if each of these tasks took a finite time to complete. If each of these tasks took $1s$, then, yes, it'd be true that completing all these tasks would take an infinite number of seconds.
But, the truth is that none of these tasks take a finite time to complete. If we divide the objective of getting from$1m$ to $2m$ into an infinite number of tasks, and let $dx$ be the infinitely small distance, then the infinite number of tasks are:
Going from 1 to 1+dx
Going from 1+dx to 1+2dx...................
$n-1$. Going from 2-2dx to 2-dx
$n$. Going from 2-dx to 2.
Well, none of these tasks take a finite time to complete. The time taken is also infinitely small. In each of these tasks, the distance we've to cover is $dx$ and hence the time taken is the infinitely small, $dt=\frac{dx}{v}$, if our speed is $v$. And, an infinite number of infinitely small times add up to give a finite time required to get from $1m$ to $2m$ like this: $$\int_1^2\frac{dx}{v}=\frac{2-1}{v}=\frac{1}{v}$$ So, I don't see any paradox. Where's the paradox?
I don't understand why people introduce concepts like quantization of space and time to explain Zeno's paradox.
Zeno's paradox is fundamentally a philosophical problem. Many philosophy problems can be approached mathematically, but you have to be careful to justify that you're using the right math. In your solution, you're choosing to model time in a certain mathematical way. That needs to be justified. You could also (as you mention) solve the problem by modeling time and space with the integers and just asserting that there's only finite units of both, so the paradox doesn't arise.
However, doing math in these models isn't a complete philosophical argument. You need to argue that time is, in fact, like the real numbers in the relevant way. Otherwise someone can just say "well, in your model your idea works but in the real world it doesn't because the world doesn't look like your model."
There are many ways in which Zeno's paradox can fail. The question is which one is the way it actually fails.