Unfortunately that's a bit outside of my area of expertise, I never really see false negatives being discussed when this comes up, I'm sure they happen, not many tests are 100% foolproof either way.
Most lead tests are really designed for paint, so using them on a pan is kind of outside of what they're made for to begin with and I'd be a little skeptical of the results either way.
My understanding, and it may be wrong, my knowledge on the tests comes mostly from places like this where people are cast iron geeks more than chemists, are that the tests are almost more of a test for the presence of metal in general not so much specifically lead, so you get a lot more false positives because of course there's more metal in a metal pan than you'd expect there to be in any kind of paint, and it's better to err on the side of caution and give a false positive instead of a false negative.
Again, that's secondhand knowledge from people who I don't have a ton of confidence in having their facts straight to begin with, so take that with a huge grain of salt.

It's been a long time since I watched the show, and I never read the manga so this is secondhand knowledge so take it for what it's worth
But IIRC the show actually picks up partway into the manga, and I think in the first chapter (volume? Issue? Not sure how it was organized) of the manga he is solving the puzzle.
I think the first part of the manga featured other games besides the card game so I suspect for the show they just skipped to the part where they could sell cards as a tie-in to get that merchandising going.
I think there was also an earlier TV show before the more popular one, but I have no idea where that picked up or ended.