Without commenting on whether this presentation matches the original metaethics sequence (with which I disagree), this summary argument seems both unsupported and unfalsifiable.

1. No evidence is given for the central claim, that humans can and are converging towards a true morality we would all agree about if only we understood more true facts.
2. We're told that people in the past disagreed with us about some moral questions, but we know more and so we changed our minds and we are right while they were wrong. But no direct evidence is given for us being more right. The only way to judge who's right in a disagreement seems to be "the one who knows more relevant facts is more right" or "the one who more honestly and deeply considered the question". This does not appear to be an objectively measurable criterion (to say the least).
3. The claim that ancients, like Roman soldiers, thought slavery was morally fine because they didn't understand how much slaves suffer is frankly preposterous. Roman soldiers (and poor Roman citizens in general) were often enslaved, and some of them were later freed (or escaped from foreign captivity). Many Romans were freedmen or their descendants - some estimate that by the late Empire, almost all Roman citizens had at least some slave ancestors. And yet somehow these people, who both knew what slavery was like and were often in personal danger of it, did not think it immoral, while white Americans in no danger of enslavement campaigned for abolition.