I see no way the shed shown in that picture is ever going to work long term.
They have the bottoms of the doors sitting flat on the patio. Water is going to just be wicked up through the rails and rot out the bottoms of the doors.
Ever see what happens to a sheds walls when someone has it sitting to close to the ground?
There's dozens of post on this site and all the DIY sites on what happens to homes when the decking is install right under the doors thresholds.
That whole thing needed to be built on a platform made of pressure treated wood to get it off the ground.
From the picture I can not see how there even able to open the doors, it looks like they would just drag on the ground. If you cut them off so it would open there's going to be a big enough gap for insects and mice to get it.
Also in most cases an int. door is made out of wood that's not rot resistant.
They also have a dead flat roof that looks like it's made of plywood. Just not going to work out over time.
When plugging old door knob holes I use plugs cut with a hole saw, mix up some Durham's Water putty instead of glue and apply it around the plugs outside edges, once it hardens You can use the same putty as a wood filler.
It's sandable, paintable, and as strong as the wood once it hardens.
It will also work to fix where the latch was.
