The words irreversibel and unumkehrbar seem to be used with about the same frequency. Duden, under Häufigkeit (irreversibel unumkehrbar), lists both as belonging in the group of the 104 to 105 most frequent words. A quick search in the Deutsches Referenzkorpus confirms this, with 8407 : 8721 hits.
Now even though the numbers may hide the fact that these words are used with different frequencies in different types of texts, they make me fairly confident in asserting that irreversibel is not babble as Janka suggests, but a pretty unremarkable word and definitely not limited to scientific discourse.
Per se, none of these words are preferable over the other. Sprachpurismus is a thing of the past, although contemporary Sprachkritik may still criticise the unnecessary use of words of foreign origin when an acceptable native word is available, as is the case here.
Janka's post is a strong testament to the negative reaction you may garner if you go against that injunction. So one could add to your quite reasonable suggestion that the speaker in your video adds unumkehrbar in order to translate irreversibel for those that don't know the word the following: that she is aware of the injunction and is, in a manner, correcting herself in order to follow it.
To turn this back to the beginning: Despite the still-present injunction against the unnecessary use of loan words, irreversibel is used as frequently as unumkehrbar, which could be taken to mean that most people do not care either way.
This post is already long enough as it is, but I wanted to introduce two further examples (or classes of examples).
ganzheitlich and holistisch are one frequency level apart. Very likely, this means that holistisch is used in more specific contexts (it may be technical language) or only by the more educated (or those wishing to appear so).
einen Anreiz schaffen and inzentivieren. The latter is so rare as to not be listed in dictionaries. It might be a technical term in the narrowest sense, i.e. a term that is only known by specialists in a certain subject. It is felt to be a lazy and unnecessary loan from English, perpetrated by people who encounter incentivise in English texts and cannot be bothered to translate the term.
Of course, with time, inzentivieren may move up to the status of holistisch and on to that of irreversibel.