My question is, do they only show the gates on the airside departure boards, and if so, why?
Yes, would say it's to encourage pax to clear security and visit the shops in the sterile area for longer. The gates are not displayed on boards in the sterile area immediately, there's usually a "gate opens at xx:xx" display to encourage pax (punters?) to stay in the sterile area longer and visit the shops.
Suspect airport management would say it's to even the flow and avoid crowding at pinch points.
You pay your money (literally) and take your choice.
It's not just at LGW.