If I was you, I will create a custom plugin ( Calendar sync ) where you can add deferent calendar connections similar to providers in Settings.where you add google with a pen icon you click on it to add authentication config
In the plugin controllers you create functions ( add/cancel/edit event on calendar ) where you loop throw the calendar providers and execute the api requests ( Google api endpoints)
and then you call this controllers in the appointments create controllers/services depending on your need
P.S: It seems interesting what you creating , are you creating booking system using Strapi as BE and reactJs FE, I am planning to do something similar soon