If the above statement is true, then, instead of defining a (real) random variable as a function from a sample space of some probability space to the reals, could we equivalently define it as a subset of ℝ associated with a CDF?

No. To one and the same CDF, there might be multiple very different associated random variables. Sure, these random variables will have the same probability distribution, but they're not the same as function.

In its current state, that article has some self-contradictions. The abstract definition it gives for random variable looks correct. The statements it makes about cumulative distributions are misleading.

Consider the following situation. We have two questionnaires. Questionnaire A is designed to rate a persons interest in applied mathematics "on a scale" from 0 to 10. Questionnaire B is designed to rate a person's interest in basketball on a scale from 0 to 10.

A realization of Random variable X is defined as "Pick a student at your school at random and administer questionnaire A. Let X be the student's score on the questionnaire. A realization of Random variable Y is defined as "Pick a student at you school at random and administer questionnaire B. Let Y be the student's score on questionnaire B.

It is possible that random variable X and random variable Y might have the same distribution and same cumulative distribution. But the two random variables are not the same random variable because they are not defined on the same probability space. X is defined on a space of events having to do with questionnaire A while Y is defined on a space of events having to do with questionnaire B.

A graduate course in probability will make this definition more precise.

The idea is that you have a sigma algebra, a Borel space and you mix the two up to make the idea of a probability space [with its events and actual probabilities] consistent with what a probability space actually is.