Ethnicity is entirely arbitrary, that's why it should be avoided.
For example, the genetic line of what your calling the clearly Native Americans traces back to Asia. You've arbitrarily decided that their ethnicity is defined at a certain time in their lineage. Ehy aren't they Asians or even Africans?
There is culture that you can tie to a group of people at a certain time and that group frequently has similar lineage. That's usually what people mean by ethnicity. They're using physical characteristics due to lineage that is associated with a culture and they call that ethnicity.
In this context of of Romans, there is really no ethnicity that should map to the cultural identity of "Ancient Roman" as the empire had many subjects across it's empire.