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.