Can someone please walk me through (or cite a reference to) exactly which fixed probability space is used in an application of the CLT, especially in the finite probability space case?

My question comes from the following: generally, the CLT is stated as "let $X_i$ be an infinite sequence of IID RVs on a probability space $\Omega$ with finite mean and variance ...".

Traditionally, a text might say something like "the $X_i$ represent a sample from a common distribution $F$" - i.e., $P(X_i<=a) = F(a)$ for each $i$ for some distribution $F:\mathbb R\to [0,1]\subset\mathbb R$.

In the case of an infinite $\Omega$, one often thinks of $\omega\in\Omega$ as an infinite sequence $\omega=(\omega_i)$ perhaps where each $\omega_i$ is a coin flip or a repeated measurement that might carry error. In that case, one often uses something like $X_i(\omega) = \omega_i$ in a "ticket" formulation of an RV (i.e., the RV "reads off" the $i$th entry on the "ticket" which represents $\omega\in\Omega$).

In the case of finite $\Omega$, it is generally said that the finite population $\mathcal P$ of individuals (say, people whose heights we are measuring or parcels whose weights we are estimating etc) is distinct from $\Omega$ and "leads to" probability spaces $\Omega_n$ in which the points $\omega\in\Omega_n$ represent possible samples of $n$ members of the population $\mathcal P$. But then, the CLT is intended to apply to a single, fixed $\Omega$ rather than a family of $\Omega_n$. So, it seems that things get slippery at this point when we are taking limits and start considering more than one $\Omega$.

I've seen in various places various assurances like "$\Omega$ doesn't really matter, just focus on the random variables and distributions that you have direct control over". Occasionally, there will be vague reference to Kolmogorov extension or Ionescu theorems, but those seem to be a lot more about general stochastic processes than rock bottom CLT applications.

I'm looking for a clear, step-by-step, unambiguous application of the $\Omega$ concept in this case so I can have seen it once to anyone's satisfaction and then not have to worry later that I'm missing something or misapplying the theorem.

PS: please note that I've posted a related but not at all identical, separate question which also stems from my amateur's attempt to understand this important theorem:

What is the rigorous justification for applying LLN or CLT to finite probability spaces?