Toolset Access has this bug https://toolset.com/errata/cannot-set-post-author-when-post-type-managed-by-access/
That bug, is probably resolved as I cannot see it happening anymore.
Yet, a new bug has been introduced, as some folks pointed out in that erratum, however, they seem to have either never reported it properly, or probably with the wrong steps.
The BUG:
When adding a new post, using Classic Editor, as an Editor role, then you cannot select Admins as the author of this new post.
Steps:
1. Create a user of role Editor.
2. In Toolset Access set the native WP Post Type "Posts" to be controlled by access, make sure editor has all possible rights
3. In Toolset Post Types, make sure the native WP Post Type "Posts" has "Classic Editor" checked as the preferred editing mode
4. Log In as said Editor, create a new WP post. Try to set the admin of the site as the author: it will not be possible.
It does not happen in blocks editing experience.
If a post already has an Admin set as author, also does not happen, no matter the editing experience.
Here is a Duplicator package: hidden link
It is not a breaking problem, however extremely annoying since most sites client do not use Blocks, and they cannot just elevate editors to admins to resolve this issue, or disable Access altogether. Thus some quick solution would be appreciated by several clients