Can waiters see tonight's bookings on their phone without asking the host?

Yes, once a booking has been given a table. A free table on the floor shows its next booking on the tile in copper, with the time, the name and the party size, from three hours before until an hour after. That is what stops a walk in being seated at 19:50 on a table promised for 20:00. Give the table on Orders → Reservations & waitlist with the Seat button or when you take the booking by hand. The Tables, tabs and the kitchen screen tab in the docs covers bookings on the floor.