I think it should have been called Sir Jack, First Part, as Falstaff towers over everybody else in King Henry IV, Part 1. See my blog on the play:
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Most Shakespeare plays have a jester, who is able to perceive certain things better than the "noble" person. There are other elements that make Falstaff more interesting, such as the juxtaposition of "fortune," class, or perhaps simply initiative.
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No "strong current of magic runs throughout the play". It's in one or two scenes in part 1.