Your second screenshot is rather misleading because the links Settings > SCORM/AICC administration > Locally assigned roles and Settings > SCORM/AICC administration > Permissions are not for enrolling users in the course, but are for assigning roles and changing permissions for the activity.
Creating a role for enrolling users in courses sounds easy, but the reality is that it's rather more complicated because you have to also allow users with the role to access the course administration pages.
I suggest you try creating such a role as follows:
1. Create a copy of the manager role by clicking the x2 icon in Settings > Site administration > Users > Permissions > Define roles
2. Assign a user the manager copy role as a system role
3. Log in as the user with the manager copy role using a different browser i.e. stay logged in as an admin in your usual browser e.g. Firefox and log in as the user with the manager copy role in a different browser e.g. Google Chrome or Internet Explorer
4. As admin (in Firefox), edit the manager copy role and gradually remove permissions by unticking the allow checkbox (no need to prevent or prohibit anything)
5. As the user with the manager copy role (in Chrome), go to an enrol users page in a course and keep refreshing the page to check that you can still enrol users whilst more permissions are removed from your role
6. Keep going back and forth between the two browser windows, removing more permissions then checking that you can still enrol users
In this way you'll determine exactly which permissions are required in addition to the enrol users permission. You will need to accept that users with the role will be able to do more than just enrol users, so you should only assign the role to users whom you can trust!
I hope you can follow my suggested method. Please let us know how you get on!