Probabilists don't care, what exactly the domain of random variables is. Here is an extreme comment that exemplifies this: "You should soon see, if you learn more stochastic stuff, that specifying the underlying probability space is a BAD IDEA (what happens when you add a new head/tails?), and quite useless.
If specifying the underlying probability space domain of a random variable (in short "domain" from now on) is such a useless, bad idea for most scenarios, I'm wondering why no one in the long history of probability theory resp. statistics has come up with a better, slicker definition of random variables, that avoids this unelegant we-have-a-domain-but-we-won't-talk-about-it situation?
It seems the only reason to keep the domain $\Omega$ is to enabling a coupling of random variables, so that we can speak of their independence. But can't such a coupling be realized in a more elegant way, than using a space that we don't want to define in the first place?
As soon as I'm reading texts that go beyond very elementary probability, it seems to me that such domains are treated like the crazy uncle from family parties: which we never show them/him, but know it's there.
That specifying a probability space is a bad idea in lots of cases does not imply that the definition of probability space lacks formal elegance. On the contrary. Exactly the fact that it can stay "undercover" while handling probability problems is in my view very handsome and somehow a proof that things cannot be made better. The quote in your question states a very recognizable fact. The modeling that goes along with solving the problem can always start with something like: "Let it be that $X,Y,Z$ are random variables on the same probability space, such that ...." What probability space? Who cares. Actually the only thing of importance is that such a space can be constructed, and that the spaces are somehow "isomorphic" when we restrict to the relevant issues. How to construct such model/space must be a part of the course on probability, but this merely to make certain that it is possible. If that knowledge has landed then we can step over on "faith". From that moment on we can just believe in it, and this in the nice certainty that we believe in something that is true. I very much like that comfort.