Missing DatabaseMail.exe.config after applying SQL Server 2016 SP2 CU1
The main point of the question was to check if other people had the same issue and to help spread the information on issues with the SQL Server 2016 SP2 CU1 patch. Since these objectives have been reached and there is a solution presented in the post, I will close this as answered.

The solution was as follows:

We recreated the file based on the following KB article:
https://support.microsoft.com/en-us/help/3186435/fix-sql-server-2016-database-mail-does-not-work-on-a-computer-that-doe

After that we had to restart the Database Mail queues to start processing queued mails:

USE msdb;
EXEC sysmail_stop_sp;
EXEC sysmail_start_sp;


This cleared the issue and any queued messages were successfully sent.

Patch link: https://support.microsoft.com/en-us/help/4135048/cumulative-update-1-for-sql-server-2016-sp2