I have done this "credit-by-examination" as a student. This was called "challenging a course".
The precise rules can vary between different schools (colleges/universities). There is commonly a maximum limit to how much can be challenged (number of credits/courses). Not all courses can be challenged; this may be up to the department (e.g., perhaps the department chair).
For instance, I was once coerced into challenging a course. This satisfied the bureaucratic requirements which were being imposed, despite the detail that the requirements were an unwritten update that didn't match the course catalog. Had I done things the way the department recommended, I would have needed to graduate 15 months later (due to when courses were made available).
I got full credit for the course, despite doing nothing but reading a book and taking an exam (and paying tuition, and filling out a form). My record simply shows that I took the course and got an A.
In no way do I suggest that I had the exact same experience as if I went through the entire course, including delivering presentations and completing the projects. Yes, the challenged course was a streamlined experience, and I am basically getting full credit for far less work.
I've received credit for some other courses by having industry certifications (which required I take an exam, which can be taken in exam centers other than colleges...) My mother has received credit for prior "work experience".
Should this be allowed? Maybe not. However, many universities do allow this. Is it unfair? Well, an argument could be made that this is fair, because the college student catalog may describe policies including the ability to challenge a course, or get credit via other means. I think that some students could reduce overall workload by exploring such options, although they are often simply unaware of the options. The educational institutions often don't go out of their way to promote such non-traditional approaches, because it involves a bit more work by staff, perhaps some more expense, and may produce graduates with a bit less experience. So they have no particular incentive to promote such familiarity. Personally, I do try to tell people about such options, because I am not a fan of people failing to benefit from available options just because they were unaware of opportunities that were technically available and even published (at least partially) in their student catalog.
I don't really think that works as it misses out on a lot of the material that was assessed via other means (e.g. paper writing, presentations, etc) as an integral part of the course.
This sounds true.
Don't expect that everybody who gets credit for the course has had identical experiences. Staff may have some ability to influence the requirements to get credit for a course, or even a degree. (I recall once when a university lost staff and was unable to teach a required course. Until the university was able to adjust the requirements, students were routinely told to just file a waiver asking for an exception, and graduating without taking the "required" course.)
The head (or other staff) of the college/university/department staff might be able to impose additional requirements. Maybe not in this case, if there is an official "credit-by-examination" process. Even if the staff is able to impose some additional requirements, they still might make decisions (different than what you might make) about what the student should (or shouldn't) be required to do, in order to receive credit. If the student gets credit for the course, or not, is really a decision that is quite separate from the task of making this exam.
While the department head suggested just using the final exam as the basis
I suggest reviewing the final exam. Then, see if there is anything major missing (maybe the final exam tends to focus on only later topics, and crucial earlier comments are entirely absent). However, don't try to use the exam to re-create aspects of the "skinny and deep approach (assessed via papers and presentations)". Just make the exam so that it fulfills what the exam is expected to do: ask questions about what is normally "assessed via tests" throughout the course.
Trying to make the exam into something unusual may be something that wasn't expected by the person who decided what a student should be required to do to satisfy the course's requirements. I suggest not trying to subvert the process by crafting an unusual exam. At least, don't spend a lot of time and effort doing that before finding out whether that would even be desirable by whomever provided you with this task.