I appreciate your request, and it might be a nice additional feature. When you think about its logic, it could be rather complex. Do you want only teachers to be able to mark a item as complete, either the teacher or student, the teacher, student, or criteria being met, etc.
I thought about your example, and offer some ideas.
"- A teacher might be more interested in the quality of the a student's posts rather than simply the quantity. This would prevent students from gaming the system."
This can already be accomplished with a rating scale and summing the scores.
"- Instead of uploading a report, a teacher might accept a physical model or prototype."
This can be accomplished by putting a grade in the grade book. Once a grade is entered, the activity is complete.
"- In fact, there could be other assignment types that are not text input or file upload: classroom presentation, field trip attendance, seminar attendance, submission of physical object, etc."
This can be accomplished by putting a grade in the grade book. Once a grade is entered, the activity is complete. I do this all the time.
"- it's also possible that for one reason or another the student was no able to take the test on Moodle so a paper test was taken instead. Assuming the student received a passing grade for that test and the score is not critical to the overall grade which is automatically calculate by Moodle, it would be sufficient to simply mark the test as completed."
In Moodle, you can have exceptions to quizzes, and you can even leave a quiz as "blank" (in case you decide to let students skip an exam.) If one wants to show something as complete (even with a score), you can put these items into their own grade book category and give it a "0" contribution to the final grade. There already is a lot of flexibility in Moodle to do this, much more than in products like Blackboard.
So even though I think one could already handle all of the examples your present, I still like your idea. I suggest adding it as a "feature improvement" in the Moodle Tracker system, trying to describe as best you can what you want this feature to do, and then encourage people to "vote" for it.