[pcrh]

>You enroll in the school for ~3 months, write the missing parts, and that's it.

There are degree mills that do what you describe.

There is also the format in countries such as Germany or the Netherlands where one typically "bundles" one's publications into a thesis. However, the work is typically done in the context of supervised doctoral programmes and no less rigorous than that done under different PhD studies formats.

[pclmulqdq]

TIL Cambridge University is a degree mill.

https://www.cambridgestudents.cam.ac.uk/exams/students/postg...

Like I said above, this is not usually offered unless you are clearly doing work of sufficient quality. 3 ArXiV preprints can get you a PhD just fine, but it won't cut it if you wrote them when nobody is watching.

[pcrh]

That's only available to those with an undergraduate degree from Cambridge who, subsequent to their undergraduate degree, publish work worthy of a Ph.D. and pass a viva.

I'm not sure how common a route ot Ph.D. it is (I never heard of it before), but it sounds like an anachronistic extension of the MA's Oxbridge graduates feel entitled to.

[pclmulqdq]

Correct. It is generally an exceptionally rare path to a PhD, but the door is open. I believe I saw the statement that this particular program has granted ~100 degrees over its life compared to ~15000 PhDs from Cambridge.