You should be able to add the respective responsibility by using the Sysadmin responsibility. As far as I know it's available by default. Can try sear searching as '%Desktop%' and confirm whether the the responsibility is unavailable. If you still don't find the responsibility can ensure the respective responsibility is not end dated.
This responsibility is not available in 12.0.6. You will need to upgrade to 12.1.2 (or above) and there is no patch available for your EBS release.
Oracle Web Applications Desktop Integrator Release Notes, Release 12.1.2 (Doc ID 950876.1)
Oracle E-Business Suite Applications Technology Readme, Release 12.1.2 (Doc ID 845809.1)
It's nothing to do with seniority, I came to know about it just today after seeing this post .. No one knows everything, everyone knows something
And, I must say I'm very impressed with your knowledge and professionalism, please keep up the good work.