The intuition behind the probability space $\Omega$ is, I think, that it is the state space of "the system" (the existence of $\Omega$ means that we assume that there actually is something that can be called "THE" all encompassing universal state space, and I think that denying thatsuch a thing is the philosophical stance of the Baysian, even if he uses it to prove Bayes law). In that intuition a random variable is some value of the state, which is random because we don't know the state of the system. We can, at best, say what the probability is of a set of states. Ideally we have some a priori symmetry principle which says that each state of the (often huge) state space is equally probable (or more generally and technically that there is some natural $\sigma$-algebra and probability measure on the state space). The pointgoal is then to determine the probability of the outcomes of a function depending on that state.
The canonical example, and the birth of probability theory by Fermat, Pascal and Huygens (http://homepages.wmich.edu/~mackey/Teaching/145/probHist.html), is determining the probability of wins and losses in gambling where. Here the state space is the set of all possible hands that can be dealt or all possible n-dice outcomes that can be rolled ,and the. The random variable is the loss or gain under the rules of the game. The probability for each state is unambiguous and relatively easy to determine whereas the probability of an outcome for the total number of points in a diceroldice rol or the points in a hand of cards requires, require the enumeration of the number of ways to realise an outcome.
I think that a lot of the more technical development of the subject came from the desire to formalise statistical mechanics and Boltzmann's principle. Here the two typical examples are the computation of magnetisation of a lattice of spins (the Ising model in d-dimensions) and kinetic gas theory both of which can be seen as application of Boltzmann's principle, which says that without extra information the probability to be in a state x is proportional to exp(-\beta E(x))$\exp(-\beta E(x))$ for some inverse temperature \beta = 1/T > 0$\beta = 1/T > 0$. T$T$ is called the temperature and equals the thermodynamic temperature in physical systems.
Since the question was about finite systems we only consider the Ising model. For the Ising model in d-dimensions we have a finite state space which is $\Omega = \{-1,1\}^\Lambda$, where $\Lambda = {\{0,1, 2,,...N\}^d}$, i.e. a "spin" $\omega(\lambda)$ with value $±1$ at each integral point $\lambda = (m_1, m_2, m_3)$ of a cubic lattice (part) $\Lambda$ with integral coordinates with $0 \le m_i \le N$ and $N \gg 0$ (in fact $N^3 \approx 10^{23}$). Then the energy of the configuration $E(\omega) = \sum_{\lambda, \mu \in \Lambda, |\lambda -\mu| = 1} \omega(\lambda)\omega(\mu)$. According to the Boltzmann principle, The probability of a state $\omega$ is given by $P(\omega) = exp(-\beta E(\Omega))/Z(\beta) $$P(\omega) = exp(-\beta E(\omega))/Z(\beta) $ where $Z(\beta)$ is a normalisation constant called the partition function. The magnetisation is then $M(\omega) = \sum_{\lambda \in \Lambda} \omega(\lambda) / N^3$. The name of the game is then, to determine the expectation value $\mathbb{E} M$ in the limit $N\to \infty$. In $d =2$ this has actually been done by Ising. Trying to make sense of a statespacestate space $\sigma$-algebra and measure in the $N\to \infty$ limit leads you directly to rather serious measure theory and the probablyprobability theory inof Gibbs-measures (https://www.math.uni-bielefeld.de/~preston/rest/gibbs/files/specifications.pdf)