What happens if we take this singular individual off the board? Sargon came up in what was already a dynamic and escalating period for Mesopotamian history. Great forces were at work, and would play out dramatically in his absence.
The region was suffering a single environmental crisis that was exacerbating a single horrendous crisis of debt. Already the administrative classes of various cities and two ethnicities seem to have been bleeding into each other at the edges. And obviously the vast majority of people were in similar positions - facing slavery, disintegration of traditional rights and families - as they struggled to survive crop failure with predatory lending their only recourse.
And a reformist solution to the primary crisis had been innovated just decades before Sargon's rise to power. The trouble was that traditional elites hated government mandated debt-cancellation, and turned on the guy who attempted it.
So you had a crippling and almost universally hated regional problem that elites openly recognized as a problem, and had a known solution for, but refused to actually solve outright, because the solution was seen to be too radical. And you had traditional sources of redress (religion) directly complicit in the economic crisis - they were the first major lenders, and probably invented usury. And you had the crisis forcing people off traditional lands (flight creating folk movements; lenders moving their debt slaves around) where they encountered foreigners across the floodplain all facing the same threat, to the point it overshadowed local issues.
A true radical movement was almost inevitably going to come to power somewhere, right? They'd cancel all debt and be universally loved, except for the traditional ruling classes, who would murder them at any opportunity and write to cousins and in-laws in neighboring cities, demanding they come murder this illegitimate council (or king).
So war is inevitable, but it's traditional elites who pay the soldiers. The only decent weapon the revolutionaries have at hand is revolution. As it ever was. So one side masses lots of little mercenary armies, who are mercenaries because they escaped debt or are hoping to buy their families' freedom. The Mesopotamian ruling classes aren't themselves militarized at this point. The other side hastily trains levée en masse and charges in to cries of "Amagi!"[1]
It could go either way, but obviously if the pretender's revolution isn't put down, it can only snowball. The socioeconomic systems are (well, appear) incompatible. There aren't outside forces able to intervene.
So suppose this movement wins. Someone's at the head of it. Could be named anything, but the traditional ruling class has been decrying his utter illegitimacy for decades by that point, while the other 90%+ of Semitic and Sumerian Mesopotamians view him as perhaps the only legitimate ruler outside legend. Perhaps the most logical assumed name under the circumstances would be along the lines of "Legitimate King". Which is Sharrum-kin, in that era's Mesopotamian Semitic. Which might be elided to something like Sharken over time. Which might end up "Sargon" if Hebrew-speakers were saying it.
So what would happen if Sargon didn't unify Mesopotamia?
Well, then Sargon would unify Mesopotamia.
[1] The first recorded word for "freedom". Like most such terms it was coined as a negation of a term for servitude, rather than being a pre-existing word. Literally "return to the mother", because families forced to give up debt slaves to their creditors perforce started with the youngest so the lost labor would be less likely to create a debt spiral.