It is, as always, more complicated than Carlin cares to make out. Alexander was the last of a long line of Greeks (let's just accept him as Greek for the moment - I'm not particularly interested in engaging the Macedonian/Greek issue) that tried to take out Persia. Aristagoras had riled the Ionians up as early as 499 with Athenian support, and conflict with Persia occurred periodically for basically the next two hundred years. Aggressive Greek generals operating against Persia include Cimon, Agelisaus, Lysander, possibly Cyrus the younger as a Spartan puppet supported by Spartan-led mercenaries. The conflict with Persia became the defining aspect of Greek self-identity (as far as Greek self-identity was a thing - they were happy to abandon pan-hellenism when Ionic or Doric identities were more convenient), and in fact it wasn't Alexander's hellenizing that he was most known for, but his adoption of Persian customs. i would rather attribute the spread of Hellenism to the surviving generals, especially the Ptolemies. The Idylls of Theocritus, as much as they represent an imaginary world, expect the reader to be familiar with an imaginary Sicily, a cosmopolitan Alexandria, mercenaries recruited from all over the place.

I don't think this was inevitable. Ptolemy might have lost to Seleucus or Perdiccas or any of the surviving generals, and in fact he came close to losing several times before the successor states stabilized. And of course in the end the Greeks weren't able to keep control of Persia.

Alexander did seem to have a cultural program, but it was more one of accommodation rather than hellenization, and he just happened to be the last Greek in a long line to try to take Persia out. If Alexander hadn't done it, the Greeks would have kept trying.

I'm afraid I don't know how well Hellenism survived in the renewed Parthian empire. I understand Greek remained in use but became debased (if that's a word you can use about a language) over time.