I’m posting Katharine’s comment about how to tell that phrase “at which point” turns a sentence into a fragment because it brings up a technique I discovered while looking for help teaching freshman composition: intuitive grammar tests.
Most native speakers, I assume, use intuitive tests from time to time. The one everyone seems to know tests whether “I” or “me” is correct in sentences like:
They’re coming with Jane and I.
The test: eliminate “Jane.”
They’re coming with I. WRONG
They’re coming with me. RIGHT
They’re coming with Jane and me. RIGHT
Turns out there are all kinds of useful tests, but nobody ever tells you what they are.
Katie’s test for “at which point”: insert a comma after “at which point” and see how it sounds.
The rules for what sorts of words can modify complete sentences seems somewhat arbitrary–i.e., not based entirely on meaning. “However” can introduce a complete sentence; “though” can’t. “At that point” can; “At which point” can’t. One way to test for this is to see if it works to pause–-or add a comma–-after the phrase in question. Cf:
“However, I did snap at friends, abruptly accessing huge depths of fury I wouldn’t have thought I possessed.” (fine)
“Though, I did snap at friends, abruptly accessing huge depths of fury I wouldn’t have thought I possessed.” (weird)
“At that point, you realize that it doesn’t express one more advantage…” (fine)
“At which point, you realize that it doesn’t express one more advantage.” (weird)
This may relate to where the intuitive ear comes in.