You have to be pretty damn anal to think using a colloquial term suddenly discredits all the other information in that article.
There is some validity to his point though. Every time I read the word "give-up-itis" I reflexively feel negative feelings towards the article. It feels as if they are trying to be funny but not in a good way.
If they had to come up with a "medical" term for "give-up-itis", I might guess it'd be "apathetic psychogenic decay", or else some Latinized phrase. https://translate.google.com/#auto/la/disease%20of%20giving%... yields "De morbo dare sursum", who knows.
I also suspect this research is early enough that it doesn't make sense to choose an official term and commit to it yet. I am in favor of people not polluting the namespace of terms with things that, upon closer examination, should have been named something else.
Choosing to continue the tradition of naming things in a way such that only a relatively small chunk of society would immediately understand their meaning is something that I've never understood.
A lot of the words with these roots are international, or nearly so:
Polish: brontozaur; Spanish: brontosaurio, Slovak: brontosaurus; French: brontosaure; German: Brontosaurus.
These kinds of words actually make English easier to learn for people coming from numerous other languages.
Sure, let's be the laughing stock and replace that with "thunder lizard" and "underheating".
Jargon has its place. In regards to medical jargon, it serves as both a signal knowledge of medicine and pointing to a specific idea. Using plain language may or may not refer to the specific idea at hand and it definitely signals a lack of knowledge.
It's the same reason people who use computers say "cursor" instead of "pointy thing".
The precision is certainly a reasonable argument, yet that may be begging the question. The reason these words are more precise than alternatives is because these words are the ones chosen to associate with whatever 'thing' we might be referencing. Blue berry is hardly precise in terms of the words used, yet you know exactly what I am referencing with those words. Had we adopted such a tradition there we might find ourselves calling them vaccinium cyanococcus in a parallel dimension, and certainly there they would likely find the notion of blueberry to be an insufficiently precise alternative.
Still I can't see how someone in the medical profession would stick the -itis suffix on something that isn't any sort of inflammation, even out of jest.
Even, say, "give-up-osis" would be an improvement; an "osis" is just a condition (neurosis, psychosis, ...).
"itis" is used in the vernacular for making up words for obsessive preoccupations or imaginary illnesses and such; that formation style is not fitting in a serious medical setting at all.