What I can't fathom is how that happens, even in principle. Isn't the assessor, and the assessor's epistemic standards, part of the context of the utterance?

So imagine you read an ancient text in which the claim "the voice of Zeus follows upon his blows" appears as a description of the relation between thunder and lightening. THe assessor and facts about the assessor are nowhere part of the context of the utterance, because the utterance was made thousands of years before you were born.

An invariantist would say 'that's false, there's no such thing as Zeus'.

A relativist would say 'That's false to me (no Zeus), but it might have been true for the person who wrote it if they had different standards for truth."

A contextualist would say 'So Zeus doesn't exist, but the voice and blows of Zeus are just terms for lightening and thunder. If someone made this comment knowing what we know about stoms, it would be false. But I assess this statement as true, because I take the utterer to be talking about lightening and thunder.'