I wouldn't want to challenge Goldsworthy's appraisal of Roman or Carthaginian warfare, but I suspect his statement is based on an outdated notion of what Greek warfare was like. Scholars have long assumed that the wars of the Greeks were deliberately limited in scope and objectives - specifically, that they were governed by rules that sought to keep civilians and settlements out of harm's way. It is believed that Greek wars were resolved in single decisive pitched battles, and that no Greek city-state actively pursued the destruction of another. However, recent research has shown that this is not the case. Though they had limited resources of manpower and money, Greeks often aimed to wipe out their enemies completely, and their enemies were therefore sometimes forced to fight to the bitter end. Also, the Greeks were famously involved in bitter wars against non-Greeks, where their resolve was put to the test.

For us, perhaps the most famous example of the willingness of Greeks to keep fighting against all odds is the story of the Athenians during Xerxes' invasion of Greece. The Persians repeatedly demanded their surrender, and even offered them very favourable terms, but they consistently refused to bow to the invader. In 480 BC, Athens itself was captured and razed to the ground; the small garrison of the Akropolis was wiped out to a man. The Athenians were left with nothing but their lives and their ships. Yet they still refused to surrender. It was the Athenians who urged the Greek coalition fleet to fight at Salamis, and forced the issue when their Spartan allies proved unwilling. In later times, the Athenians rightly claimed that they had sacrificed more than any other state for the defence of Greece.

Another excellent example would be the Spartans - not the small band with Leonidas that fought to the death at Thermopylai when Sparta was at the height of its power, but the final few hundred defenders of a Sparta stripped of its might after the crushing defeat at Leuktra in 371 BC. When the Theban general Epameinondas invaded Lakonia for the third time, in 362 BC, the Spartans had no hope left. Their lands had been ravaged, their empire lost, half their helots liberated; their enemies outnumbered them at least ten to one; no allies could come to their aid; their town had no walls for them to hide behind. Everyone thought this was the end of Sparta. But the Spartans were not done yet:

As for what happened next, one may either hold the god responsible, or one may say that nobody could withstand desperate men. For when Archidamos led the advance with not so much as a hundred men (...) and advanced uphill against the enemy, at that moment the fire-breathers, the men who had defeated the Lakedaimonians, the men who were altogether superior in numbers and were occupying the high ground too, did not withstand the attack of the troops under Archidamos, but fled.

-- Xenophon, Hellenika 7.5.12

For the Greeks themselves, though, the finest example was that of the Phokians in their war with Thessaly some years before the Persians attacked. The initial Thessalian foray was beaten off, and the Thessalians, in their fury, annihilated a Phokian picked force of 300 men and invaded again. The Phokian response became proverbial:

Their disaster created such panic among the Phokians in the camp that they actually gathered together in one spot their women, children, movable property, and also their clothes, gold, silver and images of the gods, and making a vast pyre they left in charge a force of thirty men.

These were under orders that, should the Phokians be defeated in the battle, they were first to put to death the women and the children, then to lay them like victims with the valuables on the pyre, and finally to set it alight and perish themselves, either by each other's hands or by charging the cavalry of the Thessalians. Hence all forlorn hopes are called by the Greeks "Phokian desperation".

-- Pausanias 10.1.6-7

The decision of the Phokians to kill their own women and children rather than submit to the Thessalians may be the stuff of legend; our earliest source for this war (Herodotos 8.27-28) doesn't mention it. However, it is certain that the expression "Phokian desperation" was well known. We find it in Aischines and in Polybios. Through their tale of grim determination, the Phokians had immortalised themselves.

It seems clear that even when their very existence was at stake the Greeks were no more likely to throw in the towel than the Romans. Indeed, I don't think it's quite fair to list the generals you've given as examples of a tendency to turn coat. Alkibiades was indeed a serial traitor of the worst kind - but given that he fled to Sparta due to accusations of sacrilege, his initial betrayal was not a matter of turning tides of war, but of self-preservation against the whims of his own people. Something similar may be said of Themistokles. Nikias, meanwhile, never turned coat and stayed to the bitter end as general of an expedition he had opposed from the outset, putting his duty to his state above all his personal misgivings as well as his own deteriorating health.

Meanwhile, other examples could be cited. Take the Athenian general Thrasyboulos: when Athens lost the Peloponnesian War and the democracy was replced by the bloodthirsty oligarchy of the Thirty, he fled to the countryside and led the democratic resistance. Against the thousands of hoplites and the hundreds of horsemen fielded by the Thirty, and against the full might of Sparta backing the oligarchy, Thrasyboulos could at first muster exactly 70 men. Yet he held on, and his ranks swelled with dissenters and exiles, and in the end he defeated the Thirty in battle 3 consecutive times and overthrew them.

In short, I think Goldsworthy's premise is wrong; I don't think the Romans were very unique in their willingness to battle to the end.