Is gap between the tub and the drywall necessary?
What is a mold defense board? I have done hundreds of tub surrounds / showers and have never heard of this it looks like someone may have used water resistant Sheetrock and attached the tile to that. A gap at the bottom is normal and should be filled with calking.  If there is no gap any moisture that sits on the ledge is sucked up buy the Sheetrock So no you should not extend the tile backing. You should seal your grout sealers don’t normally last more than a few years, make sure the gap from the tub to the tile is well sealed this should stop water from getting behind the tile unless there is a plumbing leak. Yes even copper may develop small pin holes and leak I have seen this a few times over the years with well water that had a heavy mineral count. Where galvanized and copper were mixed without a dielectric union, and where metal was in contact with the copper. Plumbers strapping that was galvanized being a big issue plumbers tape or strapping for copper pipe needs to be copper coated. 
So seal the grout, and the tub /tile gap and if you still have a leak it is probably in the plumbing.