You are broadly correct, apart from that last bit. Unlike the cities of Hellas proper, the Greek colonies in southern Italy were quick in developing effective cavalry forces. In Greece the emphasis on the hoplite-ideal meant that the hit-and-run tactics of cavalry were frowned upon, but presumably colonists could not afford such impractical idealism.
As warfare in Greece became more intense, the mothercities also started employing light cavalry in the "Tarentine" fashion. These would have been ordinary hippakontistoi; the actual Tarentine cavalry was still in a class of its own. The confusion is down to the usual muddle of Hellenistic military nomenclature. In any case, Hellenistic factions do get an elite skirmisher cavalry unit in the the form of the Hetairoi Aspidophoroi, so I am not sure why you would want Tarentines to become available everywhere.
[ETA: just to be clear: I think EB's Tarentines represent the cavalry of Magna Grecia in general, not just of Tarentum.]
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