I think the notion of control and the potential negative consequences of “losing/being out of control” in a live [televised] broadcast or appearance is behind the negative connotations of “partir en live” in French, this in spite of its literal counterpart in English (“going/to go live“) (merriam-webster.com) being devoid of the same pessimistic assumption that things that can go wrong will, as per “Murphy’s Law” (wikipedia.org), always go wrong [when something “goes live”].

(Although you don’t ask directly whether the expression at issue is a true anglicism, to the extent that it might help explain why your 25 years of experience with English doesn’t seem to help you in this instance [just as my 65 years’ worth didn’t help me], this difference in meaning between the French version and its English counterpart is why I would not call “partir en live” an “anglicisme” [as fr.wiktionary.org does in your question’s first link], but rather an example of “pinch[ing] and reinvent[ing] [thelocal.fr, 8th item down] or at most a “mélange de Français et d’Anglais” [originedesmots.blogspot.com].)

Back to the importance of control, both the thelocal.fr and the originedesmots links in the above parenthetical contain the notion of “losing control” in their respective discussions of “partir en live.”

Whether or not the fear of the potential negative consequences of the loss of control associated with “going live” justified the French “reinvention” of that English expression to assume the worst, the fact is that French’s “partir en live” does seem to assume the worst, even to the extent of equating it to (and making it synonymous with) the dire consequences [justifiably] associated with an airplane that is spinning/spiraling downward out of control (i.e., “qui part en vrille”).

Regarding the possible origins of “partir en live,” the originedesmots site mentions (in February 2013) that:

L’origine de cette expression est tout à fait récente, et provient de
la culture télévisuelle

and gives (from March 11, 1984 [liberation.fr]):

Un exemple de moment de télévision qui part en live est celui où
Gainsbarre brûle un billet de 500 francs devant la France toute
entière.