Why not? The last decade before semi-retirement I had all the different ways to get in touch with me restricted to my phone. My work computer had no email client, no messengers, nothing. I even helped lead the charge to eliminate desk phones.
That little display may have been the single greatest priductivity booster ever. It stayed on a shelf across the room on do not disturb. The only people allowed past the DnD were my wife and my son. If there really was a work emergency, a manager or coworker knew where to find me to tap me on the shoulder.