"cells are little computers that operate on DNA as machine code" is a really different claim from "the entire universe is fundamentally computational because Church-Turing." The article is talking about the first one.

Even if the second is true, there is still a question about whether cells are "computational" in the same way computers are (little robots programmed by dna), or whether they're computational in the same way the photosphere of the Sun is (complex, chaotic physical processes). Which of those it is matters when we try to eg assign causation to how cells operate and make predictions about them.