There are two large conference rooms in the West Wing of the White House – the Cabinet Room and the Roosevelt Room. Any large meeting in that building will be in one of those two rooms (unless the topic requires the Situation Room).
Outside of the doors to both rooms, resting on side tables, sit open faced mahogany cabinets with cubby holes a few inches wide and deep. These are for the storage of phones and similar, prior to entry into those two rooms.
One could make an argument that the meetings held in those rooms are among the most important in the world. One could further argue that the people in those meetings have a need for urgent information as great as anyone else on the planet, with few exceptions.
So if those folks can put their email, text, and calls aside while in a meeting, perhaps the rest of us can make this practice work as well?