400-year-old traces would be hard to detect owing to admixture, but if they could find identical-by-descent segments that would be very compelling, as the research into Native American traces found in Polynesian populations shows:

https://gizmodo.com/native-americans-voyaged-to-polynesia-lo...

If Croatoan ceremonies didn't involve cremation it could be quite a bit easier.. I don't really see the article's evidence as very compelling. Many things may have been collected from the site and ultimately discarded in the trash heaps without the proposed integration.

There's no descendants, bones, or other source of DNA known to belong to the colonists to work from.

The English have good records. We could perhaps find the decendents of relatives who stayed put and then find their "hey you guys seem to have more DNA in common than you ought to" counterparts of native american heritage.

The way it works for molecular phylogeny is that you try to find things that are conserved. E.g. if you find a small village in Europe where people haven't moved around much and you find a rare mutation that is also present in one other part of the US, then you might be able to put some numbers on the likelihood that this mutation/gene came from a the original place. Find a second gene, find some artefacts from the right place/time and you have an emerging picture.

The native population of the area was well mixed with European and African genes in the 18th and 19th century. It would be very difficult to determine whether there was also mixture in the late 16th / early 17th century.