After researching the error "Publishing all customizations failed." when a user tried to click the "Publish" option when exporting their solution, I found that giving them "System Customizer" role resolved it..
The issue is, that security role is too powerful for regular users.
I have a test account that I gave that security role to and found that not only could it not publish solutions without the errors. But it could also edit existing solutions in the environment and make changes to unmanaged items.
Any ideas how I could give regular users rights to click the "Publish" option when exporting their solutions without generating the error, while also preventing them from editing existing solutions?