Thank you for contacting us and I'd be happy to assist.
The fix for the missing author drop-down issue was already covered in the latest release of Toolset Access (2.8.14).
If you're still experiencing this, I'll need to see how the post type and the Access plugin are set up on your website. This will help in troubleshooting the issue on a test website.
Can you please share temporary admin login details along with the name of the post type, where the author drop-down is not showing?
Note: Your next reply will be private and making a complete backup copy is recommended before sharing the access details.
Can you please also share some information about in which post type, you'd like to see the author field and which user role(s) should be available in that field?
When I checked the edit screen for the "МИУs Онлайн" post type that you shared, I noticed that only the users with the "Administrator" role were showing in the post author field.
That is because in the settings at WP Admin -> Toolset -> Access Control -> Post Types, under the post type "МИУs Онлайн", only the "Administrator" and "Editor" roles had the "Publish" permission.
( screenshot: hidden link )
As an example, I've enabled the "Publish" permission for the "Teacher" user role too and the users with this role are now also included in the post author field for this post type. Similarly, you can also add publish permission to other user roles too, to include them in the post author field.
I hope this helps and please let me know if you need any further assistance around this.