The timestamp on the file was a machine precise measure. So it seems to me like you're wanting the benefit of the machine precision there, without the disadvantages that come with it.
I wouldn't suggest a measure where a prof was relying on a huge digital clock. I was trying to address a context that wasn't machine driven as you were advocating. There are simpler realities. A prof who will accept submissions until they leave for the day. For remote education, the physical mail thing is very real (or at least it used to be).
If neither party is benefiting from machine precision, there's no reason for the onus to be on the professor or administrator to make allowances for that imprecision. It actually makes a lot of sense for that onus to be on the student.