The marked form do/does/did ... eat is used for the negative (to help mark the 'not'), for the interrogative (to help mark the question), and to stress the utterance.
Without the marked form, we would have (and historically did have [example of stressed form]):
He ate not the cake.
Ate he the cake?
and we still do have:
Eat the cake
Eat not the cake
For negatives and questions, there must be something between the 'did' and the 'eat.' In questions, if the subject is 'who,' you could not say:
"Did who eat?" (Answer: Yes, who did.) or
"Who did who eat?" (Answer: [Mr?] who ate George.)
For stress, you can say:
"You didn't finish it."
"I did finish it."
I'm not sure that will convince your students, but I have had some (?) success with students in Kuwait using the nonsense examples.
You could look at Wh-movement - Wikipedia, the free encyclopedia