How widespread is the practice of delegating LoR writing to the applicant?
There's a spectrum of "writing your own LoR. " As a faculty member, for example, I might correspond with the student:
Asking for bullet points that the letter should hit
Asking for a few sentences or paragraphs of the highlights that the letter should address
Asking for an entire first draft that is then revised
A complete letter that is copy/pasted in its entirety
In principle, I write all of my LoRs myself. However, with students in large lecture courses or that I don't know that well (for example, those who have left for a couple of years, and suddenly need an LoR out of the blue), I might ask for a variation of #1 or #2. I always rewrite whatever they give me so that it is in my own words.
I especially ask for higlights for students who are going into industry since I've never been on the hiring side for industry (only academia) and don't know what should be accentuated .
From the student's perspective, though, when they get a #1 or #2 request - they might mistake it for a #3 request.
I've never asked for a #4 type letter and although there are rumors of faculty that do, I don't see how they serve either faculty or student purposes. Faculty who are lazy tend to also be risk adverse and signing your name to someone else's LoR is risky. It doesn't serve the student either as students don't know the genre of letter writing and are unlikely to be able to write a strong letter for themselves. So while I won't presume that in the entire universe of universities, such a case has existed, I would think the actually prevalence is quite low and the reality is #2 and #3 type requests that are misunderstood by the student to be #4s.
It's been 35 years now, but IIRC I had an undergrad professor who had me write my own recommendation. If my hazy memory is correct, I think the way he handled it was perfectly reasonable. I gave him the letter, and we may have gone through one or two drafts after that. Most likely it was much more time-consuming for him than if he had just written the letter himself. I think the idea was that if I had taken a freshman class from him, and he was writing a letter for me three years later, it would be very unlikely that he could remember enough to say much more than, "Good student, got an A in my class." Having me at least write the first draft would mean getting some more about me as an individual. I don't think there was any tendency for it to be inflated compared to a letter he would have written on his own. If anything, I think I was hesitant to overstate my own case.
These days if I have a really strong student, and I want to go the extra mile to write them the best possible letter, I usually ask them to provide me with lots of written materials to fill in my knowledge about their life. I have them send me their statement of purpose or admissions essay, and I try to draw them out by email or in person about their life, or things they did in my class that I had forgotten. In many cases I don't know until this point that they were in the military, or were the first in their family to go to college, or had had to overcome an invisible neurological disability. I doubt that the result of this process is much different than the hypothetical result I would have obtained by having them write a first draft of the letter.
I teach physics at a community college, and many of my students are poor writers, so letting the student literally write their own letter without revising it afterward would be a disaster. It would be an ineffective letter, and it would also make me look like I didn't know how to write. For the same reasons, if I have a student I really think is great, I will ask them to let me make comments on their statement of purpose or admissions essay before they send it out. Often what they give me is just abysmal, and they have no clue that it's bad. Many of our students are not native English speakers, or have grown up in households with no books. Their humanities instructors don't seem to require them to do much writing, and if they do require them to write, the standards seem to be incredibly low.
The official data on this question seem sparse. Judging by Google Scholar results, there are little to no academic publications related to the topic. One web source claims that, according to their poll, 79% of respondents have been asked to write their own letter of recommendation at some point in their lives, but the page doesn't even refer back to the original poll.
However, we can also probe the question indirectly. A Google search with the "write your own letter of recommendation" query returns 14.8 mln results, while a simple "ask for a letter of recommendation" returns 48.8 mln. A StackExchange response to a related question contains excerpts from some of those 15 mln websites, and many refer to such requests as "common" and "not unusual" (with regards to both job-related and academia-related situations). There is even a Wikihow page on writing your own letter. So, overall, it does look like a solid portion of LoR requests results in students writing the letter draft themselves.