Suppose you get into your car and start driving. You drive for 1 hour and travel 70 miles. At the end of 1 hour your car is at rest since you've reached your destination.
Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?
So I would indeed say that x cannot be any number less than, or equal to, 70. In this universe, it probably is not possible to drive exactly 70 miles per hour, because mile or hour may not be an integer multiple of a fundamental unit.
Everything would be discretized, including possible speeds. Note that actually only speed differences need to be discrete, as those are the only thing actually observable. You cannot measure any absolute speed in this universe.
In the first branch, you are assuming the existence of uncountably continuous time and space with perfect precision. Scientific observation is not on your side. (Plancks constant, QM, for example)
For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.
I don't believe I'm assuming anything. I asked specific questions. The answers to those questions are relevant to what the OP wrote. Your last sentence does not make sense to me. The universe is not a number and therefore is not a rational number. It is not rational in the sense of thought either since it does not think (as far as I can tell). I have not idea what is meant by the statement, "if the universe were somehow rational...".
Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?