I did a little digging and found that others had the same query here: Submission@assignment: Editing submission is possible although it isn't allowed MDL-43494 where the assignment maintainers confirmed that this is expected behavior if the submit button is not used, and I just confirmed in Moodle 2.8.7 that the edit button is still there for my student after I graded it if I didn't use the submit button or the cutoff date option (since I had never really tried that myself before). In this case, it may be that some of the language around what does what needs to be refined, perhaps, as it does say "submitted for grading" whether I require the submit button or not.
But the student cannot submit or change anything after I grade it if I do turn on the cutoff date whether I use the submit button or not.
Which brings me, I guess, back to the question of why this cutoff date feature is insufficient to meet your needs and why you think teachers will not understand how to use it. It is pretty standard practice in many institutions/academic departments to allow late submissions for a period of time that get huge deductions in credit if they are late and then NO credit after being so many days/weeks late.
If I am teaching a course where I am using the use-case you describe as having students turn in something at my desk that I will absolutely be tracking for timestamps/timeliness, then I also understand it is my responsibility as the instructor to be at "my desk" to receive the assignment if I am using the delayed cutoff date. This can be done a couple of ways if you use the marking workflows.
If I want Moodle to control absolute timestamps and not have to be at "my desk" so to speak, I can set the cutoff date to be the same as the due date. It isn't that much of an extra step, really. You can set your site defaults for the assignment to turn on the cutoff date by default so that teachers will need to make that decision the same way they need to make the decision to have a due date or not.