In my client instance, I am unable to see the "Admin Control" option in the inbox view. What privileges are required to see that view? I am already in the "Admins" group (with AMS). Do I need to be in "Administrator" group to see that?
The Admin Control option is visible only to the members of the administrators or workflow-administrators group. As you said you are already part of 'Admin' groups by AMS, which will a customized group created by AMS for your organization with few admin privilege such as system console access but it is definitely not part of the other 2 groups which is why you do not see the option.
You can ask AMS to include workflow-administrators into your customized Admin group and you should be able to see it afterwords.