Reread the first part of the OP's post. I do not see the term "plagiarism" in the OP's post.
The OP wrote: "I am quite sure that the work has been done by a number of people coaching her at Student Learning. If she went to someone different each time and they gave her some help, eventually between them they would have written an essay."
I teach at a small private university in the South; previously, I taught in a large university system in the Midwest. In both places, what the OP claims to suspect would clearly fall under the definition of academic dishonesty, if not plagiarism. I assume this standard isn't unconventional.
I see concern that "she will graduate with a BA - but without being able to actually write or be able to put together an essay by herself." Who does that serve?
Among the National Association of Writing Centers's operations guidelines is the following: "Tutors should be given guidelines for defining acceptable and unacceptable intervention in a student’s writing process."
My wife and I both teach college writing, though at different schools; my wife is also the director of her college's writing center. I can say with virtual certainty that no writing center advises its tutors to violate its institution's policies regarding academic dishonesty.
Regardless, if you, or the OP, or anyone else in these fora discussions you describe actually suspect that your writing centers are helping your students cheat in your classes, why in hell aren't you ringing the alarm? I know that if I had credible evidence that the writing center at my university was helping my students cheat in my classes, I'd be in my department chair's office waiting for him.
The evidence, however, that you offer to justify this concern is:
the striking contrast compared to the student's writing in class, i.e., on her own (and her inability even to satisfactorily discuss, in person, her out-of-class work).
Sure, the classic plagiarist's profile. But in my experience -- and talking with a number of colleagues at two schools in a lot of different departments more or less bears this out -- the students I've busted cheating were ones who would barely come to class. They weren't staking out my office waiting for my office hours to start so we could "spend a lot of time" conferencing about their crappy work.
From the OP's summary of events, here's what I got: the student bombed an essay exam, then bombed what read to me like an ad hoc oral exam during a conference.
(The scholarly research regarding essay exams as a pedagogical practice is contentious, to put it mildly. But that's another thread.)
So then this student -- perhaps the hardest working plagiarist in the history of higher education -- comes to the OP's office "frequently" to conference about her upcoming first assignment, an essay.
(If there are other faculty out there who's dishonest, cheating students are this conscientious, I'd like to know where they teach -- but I'll stop belaboring the point.)
Anyway, during this time, the student has been working with the OP (despite the OP's having "little expectation that she would come up with anything passable"), as well as the campus writing center on this upcoming essay. Upon reading the student's submission, the OP is surprised, not because the student has climbed out of the primordial ooze of sub-literacy to achieve the mellifluous fluency of a Carlyle, a Swift, a Dr. Johnson, but rather because her submission was "passable."
The OP wrote: "There were still a number of flaws and the approach to the question was quite a basic one but it was coherent, structured and full of relevant examples."
So, the OP's (and perhaps your) response to this student going from incoherence on an essay exam to incoherence about her incoherence during a conference/ambush oral exam to middling mediocrity on a take-home essay is -- even after extensive work with the OP and the trained tutors at the writing center -- naturally, to suspect she and those same tutors are co-conspirators to commit academic fraud.
And here's what's truly appalling about this: Exhibit A for the prosecution is the fact that the student appears to have learned too much too fast. It couldn't be because the writing assignment used to diagnose her initial "deficiency" is at fault.
I don't get the impression that you or the OP are bad people. However, I think that treating our students like de facto criminal suspects when they get help with their writing and improve is bad practice. And the idea of closing students up in a classroom to do all of their writing, just so we can watch them, is plain nutty. The fact that someone would even have to write those last two sentences on a national forum for educators is itself unbelievable.
If the purpose is to help them gain mastery, on the other hand, as you have indicated, that's certainly a laudable goal, and I don't doubt you're sincere in your desire to help your students. But there are better ways to do that. Start by sending them to your writing center.