As you state in your opening, Putin's claim "is mistaken" – i.e., a fabrication sprinkled with almost facts. Material culture in what is now Ukraine has been dated as far back as 5000 years ago. From Scythian, to Eastern Scythian Iranians, to Celts, to Turkic culture, to proto-Slav and full Slavic culture. The Rus state included Kyiv and about a quarter of what is now northwest Ukraine, but little else of what became modern Ukraine.

If we give even the slightest credence to Putin's claims, think how many other countries of which the old Rus state held corners. Are we to then give credence to claims on the entireties of those countries' sovereignty? Because, Putin will not stop at Ukraine.