It may help to think of sorites as less of a singular, concrete paradox, and more of a strategy for constructing paradoxes. For example, the Stanford Encyclopedia article about Transworld Identity describes this seemingly unrelated paradox in the field of modal logic:
One such argument, adapted from Chisholm 1967, goes as follows. Taking Adam and Noah in the actual world as our examples (and pretending, for the sake of the example, that the biblical characters are real people), then, on the plausible assumption that not all of their properties are essential to them, it seems that there is a possible world in which Adam is a little more like the actual Noah than he actually was, and Noah a little more like the actual Adam than he actually was. But if there is such a world, then it seems that there should be a further world in which Adam is yet more like the actual Noah, and Noah yet more like the actual Adam. Proceeding in this way, it looks as if we may arrive ultimately at a possible world that is exactly like the actual world, except that Adam and Noah have ‘switched roles’ (plus any further differences that follow logically from this, such as the fact that in the ‘role-switching’ world Eve is the consort of a man who plays the Adam role, but is in fact Noah). But if this can happen with Adam and Noah, then it seems that it could happen with any two actual individuals. For example, it looks as if there will be a possible world that is a duplicate of the actual world except for the fact that in this world you play the role that Queen Victoria plays in the actual world, and she plays the role that you play in the actual world (cf. Chisholm 1967, p. 83 in 1979). But this may seem intolerable. Is it really the case that Queen Victoria could have had all your actual properties (except for identity with you) while you had all of hers (except for identity with her)?
[...]
Chisholm (1967) arrives at his role-switching world by a series of steps. Thus his argument appears to rely on the combination of the transitivity of identity (across possible worlds) with the assumption that a succession of small changes can add up to a big change. And ‘Chisholm’s Paradox’ (as it is called) is sometimes regarded as relying crucially on these assumptions, suggesting that it has the form of a sorites paradox (the type of paradox that generates, from apparently impeccable assumptions, such absurd conclusions as that a man with a million hairs on his head is bald). (See, for example, Forbes 1985, Ch. 7.)
[goes on to describe a variation of the argument which is not sorites-like, by doing the whole thing in a single step]
Italics in original, boldface added for emphasis.
The broader context of this paradox is the question of whether we should identify Noah-in-world-1 with Noah-in-world-2 (i.e. whether they should be considered "the same" Noah or "different" Noahs). This paradox argues that they are different, by suggesting that we may construct the scenario such that worlds 1 and 2 are identical except that Noah and Adam have "switched places." Then Noah-in-world-1 and Adam-in-world-2 are functionally the same in every particular, despite officially being "different people." More broadly, worlds 1 and 2 are functionally the same in every particular, with only the identities of Adam and Noah differing. This is unsatisfactory for a variety of reasons. The only alternative resolution is to claim that Noah (or Adam) has some "essential Noah-ness" (resp. "Adam-ness") that does not change across possible worlds. That is unsatisfactory (or at least questionable) for an entirely different set of reasons. See the linked article for a fully contextualized discussion of these issues.
In this context, the sorites form of the paradox (in which we construct a series of worlds where Noah is progressively more like Adam and vice-versa) serves to emphasize the fuzzy edges of the standard possible world semantics. While the sorites form may be a little weaker, in that it requires additional assumptions and a more complicated construction, it is more "obviously correct," and provides a deeper intuition into why the non-sorites version is also correct. That's why Stanford leads with the sorites form of the paradox before simplifying it into the non-sorites form.
Another famous variation of the sorites paradox is the Ship of Theseus, in which a ship's planks are replaced one at a time, but it's still "the same ship." Curiously, many of these arguments seem to revolve around identity, probably because we like to think of identity as a binary, fixed relation. We are generally uncomfortable saying that A is 85% identical to B; either they are identical or they are not. This discomfort provides a fertile ground for logical inconsistency of many different forms. The sorities technique, then, can be used to build those inconsistencies up into paradoxes, which helps to define and shape the theories we construct in order to cure the underlying inconsistencies.
In this case, I fear there may be an additional misunderstanding at work. OP has, in both the question and in several comments, asserted that:
Paradoxes don't exist.
I think what the questioner means by this assertion is the following:
Flaws in reality that create contradictions don't exist.
This is, so far as we're aware, perfectly true. But it's also irrelevant, because philosophers normally define a "paradox" as a flaw in our understanding of reality, rather than a flaw in reality itself. So, for example, the sorites paradox does not mean that there is some problem with actual heaps of sand that causes them to behave oddly when we remove one grain at a time. Rather, it means that there is a problem with our definition of the word "heap." The particular definition which the questioner supplies might be one possible resolution to the paradox, but it does not obviate the paradox itself, because there could be other definitions of "heap" which might resolve the paradox in a different way.
(It would, of course, be quite absurd to assert that there are no flaws in our understanding of reality, so I have attempted to guess the questioner's true meaning. I might have guessed wrongly.)
