This is pretty much was Toynbee once fancifully imagined. It's not happening. Really. It's not.
Yet the opposite assertion, raised by some here, that Alexander was somehow doomed to see his empire crack apart... that's equally misguided. The notion that any big empire will fall apart as soon as it can is just complete nonsense. Empires bring benefits, too. Wars of succession are often the big danger. Not satraps rising up for shits and giggles. And on that note: Alexander was widely seen as a liberator, who left the local aristocrats free to run their own affairs. They had little reason to rebel. In fact, the satraps he removed and punished in OTL had been abusing their position to fleece and oppress the population, which was what Alexander did not want.
Considering his interest in decent infrastructure, his empire had a good chance to keep together for quite some time, if he'd only lived long enough to hand it over to an adult heir. It would be decentralised, but that in itself would given local aristocrats less reason to rebel.
Indeed, it's no less unlikely that Cyrus' Empire survived when he basically created the world's largest (by that time) state and incorporated multiple kingdoms and regions that had never been subject to larger authorities, yet the majority of those conquests were still under the control of Darius III at the end of the Empire. You've probably noticed I've never argued for the idea that the Argead Empire must cease to exist as soon as Alexander's dead, instead simply exploring their likely solutions to the problem of government and what issues are likely to come up in the long term.
A particular problem to these alt history scenarios is precisely that issue that's just been raised, Alexander himself being viewed as too specifically important one way or another. It is true that what he would do with those extra years, to secure his dynasty, to raise and chose his successors, to defend his territories in the long term, would have an impact on the Argead Empire's ultimate arc. But as I said before, Empires can expand, contract, expand again on multiple occasions, and not every founder of an imperial state leaves it in ideal condition afterwards. What I consider inevitable is, in a word, history; personalities and plans and accidents and desires will happen. The history of the Seleucid Empire is not defined by its establishment by Seleucus then its inevitable contraction due to the Will of History, nor that of the Achaemenids by its establishment by Cyrus followed by a series of sensuous caretakers. Individual rulers will also bring particular talents to the table, form plans, achieve new visions of their domains, and find success with the lands with which they've been endowed.
Weirdly enough I'm also arguing the same perspectives with opposite rationales; Empires are not nice. They can be beneficial to more people than their prior situation, prosperous, and safe. But interacting with an Empire is not nice, what they must do to survive is not nice, what they require to sustain themselves is not nice. That idea of a benevolent all-star union of the entire classical world under Alexander's vision is a fantasy that has lasted through the centuries, millenia even, but it fundamentally misunderstands the nature of the beast. This sort of idea is what a lot of people are refuting by saying Alexander's Empire would inevitably have crumbled immediately, because it's so clearly fantastical and easy to reject. What you and I, and others, are arguing for is an actual state that could have been formed by his dynasty.
Why should Carthage be a stretch?
Agathocles of Syracuse invaded Africa not long after Alexander's death, and got all the way to the City before being turned back. Even after his failure he got a peace of Status Quo Ante. If a pipsqueak state like Syracuse could manage that, Alexander can surely do more with his vastly greater resources. In particular, in the Sicilian War Carthage could be resupplied by sea, something which Alexander's fleet can surely prevent.
Calling Syracuse a pipsqueak makes sense in comparison to the Achaemenid Empire, say, but outside of massive imperial states Syracuse was probably the most powerful of any Greek poleis across much of the Classical era, in this era once again ruling over most of the Hellenic-ruled parts of the island. Arguing for greater resources equalling automatic victory is bizarre when we have actual history to indicate that this relationship is not so obvious; the Achaemenids were entirely capable of being resisted by, and defeated by, a coalition of Greek powers at sea, none of which at the time had a powerful maritime Empire, and were eventually defeated on land and driven entirely out of their new conquests. Yes, that land victory came after Xerxes and the majority of his forces had returned homeland, that's not cheating, that's strategy, and this is always going to be a factor in campaigns by big Empires against regions on their periphery. This coalition represented a minority of Greeks, and by no means all of the major military Hellenic poleis at the time either, some of which were aligned with Persia, some had already been overcome (Euboia never seems to have recovered from the Persians), and some declined to participate. Ancient states with massive resources can still ultimately lose campaigns to those of a smaller scale. I'm certainly not arguing that Carthage was impossible to overcome by Alexander but there is no inevitability in his overcoming it either, particularly for a state with no infrastructural links to the rest of his domains and which demands time and attention at great distance from important parts of the Empire. The Argead Empire cannot, in my opinion, be a Mediterranean and massive Near Eastern Empire at the same time.