If you look in custom properties you will find the setting
"Launching applications and unsafe files"
- For Trusted sites this is set to "prompt"
- For Local Intranet this is set to "Allow"
If your SharePoint site is internal - you should be able to add the URL to intranet zone and achieve the result you are looking for.
Edit
The "Internet Settings" are not just for IE. You will notice Chrome honors those and most application take configuration from there. These settings affect you machine wide.
When you connect to SharePoint by mapped drive it doesn't work like a normal UNC share. What windows is doing is using WebDav in the background to connect to a URL.
If you want to validate this - get a program like Fiddler or Wireshark. When you open the Excel you will see the web traffic being passed to access the file.
Protected View settings explained
- Enable Protected View for files that fail validation File
validation scans file for security problems that can result from
changes in the file structure. Clearing this option doesn’t turn off
file validation. Instead, an error message appears for files that
fail the validation process.
- Enable Protected View for files originating from the Internet The
Internet is considered an unsafe location because of its many
opportunities for malicious intent.
- Enable Protected View for files that are located in potentially
unsafe locations This refers to folders on your computer or
network that are considered unsafe, such as the Temporary Internet
folder or other folders assigned by your administrator.
- Enable Protected View for Outlook attachments Attachments in
emails can come from unreliable or unknown sources.
- Enable Data Execution Prevention (DEP) mode You can learn more in
Why is my add-in crashing?.
I've highlighted the actual rules that are probably affecting you.
The first one is what the Internet Options usually addresses. This happens because the Attachment Execution Services (AES) adds zone information to files. If a file’s zone information indicates that the file originated from an untrusted website or the Internet, the downloaded file opens in Protected View (again this is why it's not an IE only setting, AES is an OS service).
When you open the library in the browser check to see which zone it is in. If you open in IE you can do this by right clicking in a blank area and select properties. Be sure it's listed in the correct zone.

The second one is corrected by adding the trusted location into the Office Applications. You stated you had already done that but I would review it again to be sure (this is done in Office, not SharePoint which does include a trusted file location as well, but they are different) - along with the site being added to the intranet zone of your internet settings. NOTE - you can't add web address here, and because this is WebDav I'm skeptical this is the problem (but can't rule it out either).