Yes, we're trying to create a new role, but for testing, I started from scratch with the legacy roles. If I duplicate the non-editing teacher role and then allow the manage activities capability, the user sees Update and Delete icons next to each activity, but not the Move icon. With the legacy teacher role, they see the Move icon. I'm trying to avoid allowing additional capabilities one at a time to see which one enables the Move icon, but I suppose that's what I might have to do?
For a non-editing teacher, or any other role to be able to move activities you need to set both the Manage activities (moodle/course:manageactivities) and Update course settings (moodle/course:update) to allow. This may be more access than you want them to have as they now have the settings option in the admin block. You can lock down a few of the items in this area but it still may be more than you want to allow.