As a self-styled knitpicker and guardian of language, I had to look it up. The references I found [1] [2] indicated emergent can mean calling for immediate action or urgent, however, it's only in the weakest use of the word. And I find the adverb form of the word another step weaker.
I believe the definition you're highlighting is related to the dictionary definitions referring to "appearing casually or unexpectedly," such as emergent phenomena, which are, arguably slow. These near opposite meanings of the word are unfortunate, b/c people don't really know what sentences using the word are supposed to mean: Is an emergent page one that appeared gradually over time or one that must be attended to immediately?
I find the author's repeated use of the word clumsy and non-descriptive. Exactly the kind of trendy abuse of a word to signal self-entitled and undeserved superiority. To get paged emergently doesn't describe the process of getting paged for an urgent issue, since the action of paging is different from the issue (how was the page button pressed?). To "emergently reverse" blood thinners could just as easily have been stated as "quickly, immediately, or urgently."
Therefore, I would follow your suggestion and drop use of the word as a high-falootin' synonym of "urgently."