I'm thinking of failing students who didn't submit an assignment on time, isn't this the way to do it?
If you've explicitly made it clear that the assignment won't be accepted past the deadline, that's your right as a teacher.
Hopefully they learn something from my decision so next time they won't be late. Maybe.
It is proper to fail people who fail to follow the rules and turn things in on time. You should also warn them that you are very serious before the deadline so they can get it done if they haven't yet.
If I just warn them then they won't take me seriously.
You have the right to give an F if they didn't do the assignment on time. Warning them is somewhat nicer than just giving the failing grade. Help them find a way to study more and make sure their grade goes up.