Does “entsetzlich” have a stronger negative connotation than “terribly” in English? In English, “terribly” is often just a slightly stronger version of “very” and it doesn’t carry much weight.
Yes, but the original meaning of “terribly” in English also had very negative connotations. It wasn’t until the early 1800s or so that it became synonymous with “very” in terms of being a generic strengthener. See Stack Exchange for a very good breakdown of the history of “terribly”: meaning - How did "terribly" come to mean "very" or "exceptionally"? - English Language & Usage Stack Exchange
So yes, this Clozemaster sentence seems like something that was literally translated from modern English, but which ends up meaning something different in German, because (I think) the German word “entsetzlich” is closer to the pre-1800s meaning of the word “terribly”. i.e. it means something more like “disgustingly” or “awfully” or “horribly”.
Maybe “entsetzlich wichtig” is not as common as “terribly important” but I think the usage is quite similar.
A student might ask me:
“Kann ich bei Ihnen noch diese Woche einen Termin bekommen? Es ist entsetzlich wichtig.”
and I would understand it exactly as:
“Can I have an appointment with you this week? It’s terribly important.”
But in another context:
“Hast Du von Franks Unfall gehört?” “Ja, das ist entsetzlich.”
“Did you hear about Frank’s accident?” “Yes, that’s horrible.”
In both cases, “fürchterlich” might be used likewise.
In German, people using such words a bit too often might come across as theatrical.
My impression is that at least in the US it is not the case in English.