MESQUITE, Texas—In the sticky Southern heat, hundreds of workers streamed in for the 11 a.m. shift last month at United Parcel Service Inc.’s local package-sorting facility, one of dozens nationwide that help it move millions of parcels daily.

In a windowless room, a 30-year-old analog control panel about the size of a chest freezer monitors operations, with rows of green and red lights indicating when something goes awry in the building’s web of conveyor belts.

“Thirty...