This is a super cool idea, @Hayaibookusu!

I think in some ways, this could really smooth out a process and prevent tasks from falling through the cracks. On the other hand, I could see how this could 1. look unprofessional to customers (especially in a restaurant setting) 2. distract employees from tasks and could lead to personal use of their phone. 

Perhaps if the app is strictly used manager-to-manager, manager-to-lead, lead-to-lead, etc. on a work only device (like a tablet, which looks more professional and less personal).

An employee could have access and the ability to check on tasks throughout their shift by keeping a work only device somewhere accessible to all. 

This way, an employee won't get distracted every few moments by notifications, but if a lead or manager verbally assigned a task to them, they could check on the tablet if they forgot.