I got an email from American Express sharing that they are introducing a new reservation booking tool in the Mobile App. The new feature will allow Platinum Card, Business Platinum card and Centurion Members to search, book, and manage reservations from more than 10,000 restaurants (and they say growing) worldwide from the Amex Mobile App.
They go on to say that Card Members will get access to exclusive tables at hundreds of restaurants across the globe. These are the same reservations that have been accessible by contacting the Concierge service. The restaurant reservation tool within the Amex mobile app should allow you to bypass the concierge.
Note, it is not available to all Platinum cardholders yet, but it is supposed to be in early 2020.