The expected utility theorem (EUT), first and foremost, establishes a utility representation of the preference over lotteries. This is akin to establishing utility representation of preference over deterministic consumption bundles in consumer theory. The representation (in both cases) is valuable because it gives us tools like algebra and calculus to do further analysis.
Secondly, the EUT provides a very specific functional form, i.e. the expected utility form (Def 6.B.5), that is linear in probabilities. This result is due mainly to the independence axiom (see Step 5 in MWG's proof).
In terms of the use of the EUT, rather than thinking about how each axiom (e.g. independence) is used in evaluating lotteries, I would view all the axioms as a package. When given two lotteries $L$ and $L'$, I would put them through the expected utility function $U(\cdot)$ that is given by the EUT and rank them based on the output of $U(\cdot)$. This is the same as using ordinary utility functions to assess the desirability of different consumption bundles in consumer theory.
Despite the preceding paragraph, it is true that the independence axiom plays a more important role both in delivering the EUT result and in limiting its applicability.
As you noted in the comments, independence restricts the set of indifference curves to be both linear and parallel. This has been shown to be inconsistent with a lot of empirically observed behavior, most notably the Allais paradox. Much of what we know as behavioral economics started as attempts to address inconsistencies of this sort. I would point you to Burghart (2020), which decomposes independence into a part that gives linearity (which the author calls "homotheticity") and another part that gives parallelism ("betweenness"), and shows that these two parts are necessary and sufficient for independence.
Coming back to its roots, the significance of the independence axiom lies in its implications for game theory. For example, Crawford (1990) shows that, without independence as a property of the underlying preference, Nash equilibrium may fail to exist.