UPDATE - 9/12/18: See solution below
When opening a workbook in SharePoint using Excel 2016, it opens as read-only with the following message:
We opened this workbook read-only from the server.
To edit the workbook, click on the button Edit Workbook
This is a design change from Excel 2013, which opened workbooks directly into edit mode by default. The reason for the change in Excel 2016 is to prevent locking out users from editing the file.
When you open a file in edit mode, you get exclusive editing capability in the file, and no other person can then open the file to edit. They can only view the spreadsheet in read-only mode. Even if you are the only person that uses the workbook, this can happen when you open a file in edit mode and leave it open on your work machine, for example, which means you are locked out of the file for editing on another computer at home or elsewhere.
Changing the behavior to open workbooks by default in read-only mode helps prevent these scenarios. Many users read information but never make edits to the spreadsheet. For these users, opening the file as read-only works just fine for them, and it leaves the file unlocked for editing for someone else. So this change in behavior helps keep the workbook available for those people who need to edit.
The behavior has changed back to workbooks opening directly into edit mode with Office 2016 Click-to-Run Monthly Channel (Current Channel) Version 1707 (Build 8326.2058) and later. This build was released on July 27, 2017 with a new co-authoring feature, allowing multiple users to edit the workbook simultaneously. Because of this feature, workbooks no longer open in read-only mode by default. If you are on Monthly Channel, make sure it is updated to Version 1707 (Build 8326.2058) to take advantage of co-authoring and to allow workbooks to open in edit mode. If you are on Semi-Annual Channel Targeted, you will need to be updated to Version 1808 (Build 10730.20102). If you are on Semi-Annual Channel, this feature is not available at the moment and will be released in the near future.