Neither of those things proves that changes are nonrandom. I have not found a single example of nonrandom mutation in the article, just learned about new ways DNA might mutate in living organisms.
You are right that at certain points randomness is emphasised, but I think the main claim of the article is that evolution is not driven purely by random mutations.
The generally accepted theory is that progress is driven by small, continuously occurring changes (mutations), which happen solely due to DNA/RNA transcription errors (random = not influenced by external factors), and this gradually leads to the formation of new capabilities/species.
Instead the article suggests that 1. there are multiple other paths how the DNA is modified, not just by point errors (not just mutations) - examples of this are the TE changes you quoted, but also completely absorbing long pieces of DNA (and potentially entirely new capabilities) from other species, inbreeding, etc. 2. and that external factors influence the frequency of this happening, e.g. by reacting to stress (the process is not purely random)
First thing - Darwin did not knew about DNA. So new discoveries how DNA changes does cannot be used to disprove his work.
To prove that Darwin was wrong you need to find that some stress condition in some population will result in genetic changes that "counter" that condition. So "correct" (not random) DNA change is triggered by some condition.
