It's been 12 hours and I only skimmed the video, so I might be misremembering what he said. My understanding was this: Mk1 is welded from plates so at each layer of the stack you have n-1 welds holding the n plates together. My understanding of his comment was that the steel coils will wrap around the major axis of the rocket but instead of each layer containing n-1 welds to join n plates, there will just be one weld in each layer to join the beginning and end of that coil. That would still leave you with welds joining the different layers of coils but only one "seam" weld along the rocket's major axis.
No real indication he means a spiral.
There are a lot of problems with a spiral. Thin sheets of steel, even hot rolled, are very affected by the rolling process. There's also the weld line going up in a spiral. That will cause uneven heat diffusion and expansion, which is a recipe for disaster given that the skin will be cold enough to liquify helium on one side and boiling hot on the other. Then red hot during reentry. Straight lines don't have the same problems.