I thought of something else. You could get a bit of fencing or some thing like a dog exercise pen and set it up in front of the gate, on the inside, so that the chickens can't rush the gate, the person can come inside, then move the "inner fence" aside, go get eggs, and when leaving, shoo the chickens away an set the inner gate back, to exit. Is this making sense? Like a double gated area.
I was thinking this too, but hard to explain without seeing the OPs setup. Even a baby gate, something that when the main door is open a human could step over to get into the run, but that would prevent chickens from easily running out.