Nobody thinks the poems were composed in 1270. The scribe was copying, the language contains archaic forms, and several poems are quoted or presupposed by writers working earlier. Beyond that the dating is contested in a way that has not resolved in a century and a half of argument.
The traditional position, still defended, places much of the mythological material in the pagan period before Iceland's conversion around the year 1000, some of it composed in Norway before the settlement, and treats the poems as oral compositions transmitted with relative fidelity. Linguistic arguments have been used to support this, notably the behaviour of certain lost consonants that affect the metre.
The sceptical position holds that oral transmission does not preserve wording, that the poems as written down are the product of thirteenth-century Christian Iceland whatever their raw material, and that apparently pagan content is no guarantee of pagan date, Icelanders remained interested in the old gods long after they stopped worshipping them, and antiquarian composition is possible. Some scholars have argued for specific poems, Voluspa among them, being influenced by Christian eschatology; others regard the resemblances as coincidental or as evidence in the other direction.
The honest summary is that individual poems must be dated individually, that the evidence available, metre, vocabulary, allusions in datable skaldic verse, is thin and admits of more than one reading, and that no consensus exists. What is not in doubt is that a modern reader meets this material through the filter of Christian scribes writing at least two and a half centuries after the conversion.