Where are you seeing he used drywall screws? I use deck and construction screws for framing all the time, it's fine. You can always add nails too. 24" centers are ok for this too, there's very little weight. Far less well constructed sheds are moved daily.

Many sheds are built this way. Some even lighter - e.g. the overpriced wood sheds you buy pre-built use very minimal framing. And they all can be moved just fine. I commented because it's somewhat surprising how easy it is and how well it works.