Are you this boss of this admin?

If you are I would change his job description to 'stop arguing'.

Does it matter where this role sits? What matters is that it is done and someone is either willing to take responsibility for this, because theay are a valuable employee, or they are not willing to accept responsibility.

If they wont accept the responsibility for this, find someone who will.

If you are not the boss of this admin, I would suggest that you accept responsibility for this role and make sure that everyone knows you are doing this for the good of your business, since you are a valuable employee.

Steve