I think i explained twice using the race analogy.
That was not a good explanation, since the analogy is kinda off.
And even if it were not off, your conclusion in such analogy is still wrong.
In fact I could come up with a simple counter-example in which an infinite amount of entities is moving, and each of them had to wait a little (but greater than zero) amount of time after another entity has moved.
All of that within a LIMITED amount of time.
Let's name our moving entities in sequence, like E0, E1, E2, E3, etc...
Let's say that E0 in order to move had to wait 1 second after E1 has moved.
And E1 had to wait 1/2 seconds after E2.
And E2 had to wait 1/4 seconds after E3... and so on.
That is a very simple geometric series.
A bit of knowledge in calculus will tell you that if time is non-discrete and if E0 is moving, then an INFINITE amount of Es has moved since 2 seconds ago.
And, as far as we know, time might as well be non-discrete or even continuous.
There is no proof that time is discrete.