The solution was to upgrade Blender from 2.82. 2.83 had big improvements in skinning for the glTF importer.
By the way, if you ever have issues with how a skinned mesh looks in edit mode, you can also try toggling the "Guess Original Bind Pose" option on the glTF import screen.
To me it looks like the face normals are facing different ways. Game engines use backface culling to only draw one side of a polygon.
In edit mode select all A and use ⎈ CtrlN or ⎈ Ctrl⇧ ShiftN to set the normals to the same direction.
You may also want to look at not using backface culling so that both sides of the robe are visible in ...