Realtime Patient Queue Booking App

I already run my public-facing site on Google Sites and now need a lightweight queue-booking tool so patients can reserve a slot and watch their place update live. I would like you to build the entire workflow with nothing more than Google Sheets and Google Apps Script, keeping everything inside the free Google Workspace tier.

Core flow
• A patient opens an embedded form or mini-app inside my Google Site, enters a phone number for one-time verification, chooses an available slot and confirms.
• Their entry is written to a Google Sheet that acts as the single source of truth for the queue.
• On a separate “Now Serving” screen (also embedded), every patient instantly sees real-time updates to position and estimated wait through in-app pop-ups or dynamic page refresh—no emails or SMS.
• Staff can mark a patient as served or delete a no-show; the sheet and the live view must sync immediately.

Key requirements
• Phone number verification (no email login) handled entirely with Apps Script or a free service you wire in.
• Real-time UI updates powered by Apps Script triggers, WebSockets alternatives such as Google’s Realtime API, or efficient polling—whichever keeps it snappy without extra costs.
• Clean, minimal interface that fits the existing Google Sites layout.
• All code, triggers and documentation delivered so I can maintain or duplicate the solution myself.

Please outline your approach to the phone-verification step and the method you’ll use to push instant updates before we start.