Yes. To have/eat breakfast while reading the newspaper / morning paper.

When you have two actions happening at the same time, that is the classic scenario for use of the present progressive... Two things happening at the same time... I was talking on the phone while my husband was making dinner. Another classic scenario is action happening before, during and after another action happened (present progressive + simple past) I was talking on the phone when I daughter ran in. ... ... ... Your sentence above is primarily about the "habitual" nature of the present simple. That is, eating while reading is your habit, right? So... I eat breakfast (usually) while reading the newspaper. A listener would tune in to the first part of that as the key piece of info -- this sentence is about his breakfast habits. The -ing in the second part just tells the listener that the second action (the reading) is seen by the speaker as an action IN PROGRESS around (before, during and after) the eating. Actually, you probably finish eating and keep reading for a bit, so actually the eating ends while the reading goes on ... but what I said above is the theory behind how those tenses/aspect are used.

You are correct. You could say it as Most people like to eat breakfast while they reading the morning paper.