1) Is DNA "computation" turing-complete?
2) Can DNA make the basic digital logic gates, AND, NOT, NOR, etc.?
If neither of those are true, then this is an incredibly silly article and I'm really impressed.
There is mind-blowing paper, that gives exhaustive anwser to this question: https://hal.inria.fr/hal-01519828/document
This paper is about chemical reaction networks, but in 2010 it was shown, that any chemical reaction network is realizable in DNA: https://www.pnas.org/content/pnas/107/12/5393.full.pdf
2) Yes, it was figured out in 2006, how to do logic gates in DNA: https://science.sciencemag.org/content/314/5805/1585
I wouldn't be so quick to make that call. That we don't know how to do it does not mean it can't be done and there is some weak evidence (innate knowledge) that nature has indeed found a way to do this, just that we don't understand it.
I still feel this way although after a lot of googling it's a little better, I wish I had a good book or other resource on it.