I'm sure that some John Ford auteurists could make a case for it, but I found When Willie Comes Marching Home close to insufferable. As others have said, the material was the sort of lightweight stuff that could have been mined for comedy, but it's aggressively sincere to a fault. And since we're talking story here, I found the plot beyond flimsy, with so much of the not-even-90 minute film flailing around aimlessly.
It's possible that time has simply made the film seem less fresh, but The Gunfighter is the kind of movie where I'd say it has virtues almost in spite of its story. I think there's some thoughtful writing from scene to scene, and a pleasing sense of fatalism that gives the film a mature quality -- it's clearly striving for something meaningful, and succeeds to a certain extent. My issue, though, is that what I thought was going to happen DID happen, and I find it difficult to give a story prize to a film with a plot this rudimentary.
Mystery Street definitely plays a bit like CSI: Truman Era, though the film's depiction of early forensics is certainly compelling from a historical standpoint. I found the narrative to be a generally engaging crime drama, though without much to elevate it thematically or emotionally beyond B-movie territory.
My vote comes down to the remaining two films, and I don't have a terribly strong opinion either way -- both are enjoyable but not top-tier movies. And in both cases I might say the direction is even more impressive than the writing. Bitter Rice, for instance, creates a very unique atmosphere -- I don't think I've seen anything quite like the world of the Italian rice fields here in other movies -- and has some dazzling tracking shots. The material is interesting, a blend of crime and social issue film that tackles some thoughtful class & gender dynamics along the way. But ultimately the nuts and bolts of the plot are a bit simple.
Panic in the Streets, too, benefits enormously from the atmosphere -- the location shooting in New Orleans and the shadowy photography give the film much of its sense of intrigue. But the storyline, if not wildly inventive, is consistently gripping, particularly as the Widmark/Douglas follow the clues and ultimately close in on finding their targets. I also think the character dynamics in both storylines are interesting -- with the cop and the doctor having conflicting ideas about how to proceed (in both cases, for understandable reasons), and the cabal of criminals having their own internal fights over how best to avoid detection. This strikes me as the strongest narrative of the bunch, and gets my vote in this poll.