It would be sort of additional duty but he only requirement would be that you carry the gun while going about your normal work day.
Some have suggested that ALL personnel be allowed access to guns 24/7/365. This is a terrible idea mainly because while our soldiers, sailors, and Marines are brave, they aren’t angels and often do stupid things (drink until they pass out, DUI, do drugs, fight, steal, run around with other soldier’s wives, etc.) when off duty. I know this for a fact because I WAS a young sailor who did some on the list and saw others do more.
A quater century of Active Duty compels me to endorse your assessment.
Additional duty, yes. When I was deployed to Uzbekistan, a much safer place than neighboring Afghanistan, a bomb blast in the latter obliged us in the UZ base to carry our M-16 and two magazines. At work, in the mess tent, the shower, chapel, didn’t matter. Went on like this for months.
But IMO those tasked for additional guard duty need to be stationed in & around targets of opportunity like the staging area where the goblin Hasan knew his targets were concentrated and unarmed.
Issue weapons to all? Negatron. The unit first sergeants can best determine who’s most trustworthy.