You're right, the CLT and such "soft" asymptotic theorems don't apply to a finite probability space, per se. They are inherently statements about an infinite sequence of independent random variables, and as you point out, in a finite probability space, no such sequence exists. So, when someone is "applying the CLT" on such a space, normally one of two things is happening:
They are using the CLT just as a heuristic, and are not truly making any mathematical claim;
They are actually applying some theorem other than the classical abstract CLT. The phrase "central limit theorem" is often used generically to refer to any theorem of a similar form.
Let's do an example. Suppose I roll $n=50$ fair 6-sided dice, letting $S$ be their sum, and I want to say something about $P(S \le 360)$. Clearly I can define the independent random variables $X_1, \dots, X_{50}$ on a finite probability space with $6^{50}$ outcomes.
Now I might write (ignore my careless rounding):
Let $\mu = E[X_i] = 3.5$ and $\sigma^2 = \operatorname{Var}(X_i) = 2.917$. Set $Z = (S-n\mu) / \sigma\sqrt{n}$, so that $P(S \le 375) = P(Z \le 0.82)$. By the central limit theorem, we have $Z \to N(0,1)$ in distribution, so $P(Z \le 0.82) \approx \Phi(0.82) = 0.794$. Thus $P( S \le 375)$ is approximately 0.794.
This is very familiar looking, but in fact I have not done any mathematics. Indeed, merely saying that $P( S \le 375)$ is "approximately" 0.794, is a statement with no rigorous mathematical content. I have a heuristic that suggests a value that might be close, for some unknown sense of the word "close", but I have not asserted anything certain about $P(S \le 375)$.
So this is an example of #1 above. True, the hypotheses of the CLT don't apply to this setting, but I didn't actually claim that the conclusion applies either: mathematically, I didn't really claim anything at all. So as far as math is concerned, no harm, no foul.
To get an actual result, we need a "hard" theorem providing explicit error bounds. Berry-Esseen is exactly such a theorem. It tells us that
$$|P(S \le 360) - \Phi(0.82)| \le \frac{C \rho}{\sigma^3 \sqrt{n}}$$
where $\rho = E|X_i - \mu|^3 = 6.375$ and $C$ is a constant for which it is known that $C \le 0.5$. Therefore, we learn that
$$|P(S \le 360) - \Phi(0.82)| \le \frac{0.5 \cdot 6.375}{(2.917)^{3/2} \sqrt{50}} = 0.09$$
which is to say that
$$0.703 \le P(S \le 360) \le 0.884.$$
Ahh. Now that is a meaningful mathematical statement. This is an example of #2.
One thing that might be worrying you is that if you look carefully at Wikipedia's statement, it starts out "if $X_1, X_2, \dots$ are iid random variables...". Indeed, we don't have an infinite iid sequence, so strictly speaking we cannot apply this theorem. But this is actually just the Wikipedia editor being sloppy. They should have written "if $X_1, \dots, X_n$ are iid random variables...", because in fact that is the only hypothesis that the proof uses.
Even if somehow the proof does assume an entire infinite iid sequence, which I doubt, there would be a workaround. Suppose we have a proof of "if $Y_1,Y_2, \dots$ are iid, then the Berry-Esseen bound holds for $(Y_1 + \dots + Y_n)/n$". Suppose $X_1, \dots, X_n$ are iid random variables on some probability space $(\Omega, P)$ (possibly finite). Let $(\Omega', P')$ be another probability space on which we have an infinite sequence of iid random variables $X_1', X_2', \dots$, having the same distribution as the $X_i$. (You can construct such an $(\Omega', P')$ using Lebesgue measure, or with the Kolmogorov extension theorem if you prefer.) Consider the product probability space $(\Omega \times \Omega', P \otimes P')$; the random variables $X_i, X_i'$ extend to this space in a natural way. Apply the theorem on this probability space, taking $Y_1 = X_1, \dots, Y_n = X_n, Y_{n+1} = X_1', \dots$. Obtain a conclusion about $(Y_1 + \dots + Y_n)/n$. This is in fact a conclusion about $(X_1 + \dots + X_n)/n$, which makes sense as a random variable on $\Omega$.
This trick of "extending the probability space" is occasionally useful in other settings. Any time you are trying to prove something about some random variables $X_i$ on a probability space, if it would be helpful within the proof to have another infinite family $X_i'$ of random variables with any particular joint distribution, you can assume without loss of generality that they too exist on the given space - just so long as your conclusion doesn't mention the $X_i'$. It will remain a valid statement about the $X_i$.
