How would you handle this at a GPT
Note the phrase “exert an aura of integrity.” Exert is an active word… it's not just “have an aura.” If you haven't unlocked that skill, then you have to put more work into other actions to to demonstrate a proper concern for integrity. A judge's standing and reputation in a wider community (not just that store), their willingness and ability to be an ambassador to the whole community (not just that store's), and their attention to running quality events is all part of a judge's reputation, and that reputation is often the basis for one's aura of integrity.
If a judge operates only in one location and never visits other locations or participates in citywide or regional events, then yes, players outside that store might see that guy as just “the guy who's judging.” If you work events outside the store or at least play at those events and are friendly towards the judges of those events, then people coming to your local store will begin to see you as “the judge.” If players see you at bigger event or see you as someone who has helped their local judges improve in their stores, then when they spot you at your local store, they'll see you as “The Judge.”
If it sounds an awful lot like the work it takes to project integrity often has very little to do with the event at hand, you're right. How you manage yourself and your involvement in the community is an investment you make over month and years. You back it up by having appropriate levels of concern at events. Sometimes that means being very hands on with detailed work, and sometimes that means letting players see that you're comfortable with appropriate people pitching in.
Let's take the original scenario: an employee playing in the event enters a result in WER. If you block off access to the computer, that sends a message that you're either concerned. Players may think you have general concern for the event, or they may think you're worried the employee is shifty or incompetent. If instead you note the entry and say to the employee, “Thanks for taking care of that while I handled this call / deck check / break,” then you're showing that you're keeping an eye on things and that the event is ok. If you say to the employee, “Thanks for taking care of that while I sat on my butt,” then you're probably doing it wrong.
If you as a judge have enough authority, then it's ok for an employee to be given a small piece of that authority under your auspices. It's better that you have authority and can delegate than be authoritarian an unable to allow others to assist.