Hadn't heard the flap story, but the piece does get one thing wrong...Stewart wasn't directed to fly that mission, he was actually on more of a 2-week dog & pony show. He wanted to go on a mission and the USAF allowed him to go.
I don't think he was qualified on the B-52 at all.
There is also a film he made for the USAF, touting the B-58, another plane that the film inferred he flew, but there are no records of him doing so.
AFAIK, he was qualified on the B-24, B-36 and B-47. The B-36 was a big airplane, bigger than a B-52. In civilian life, he had several aircraft, the most famous being a P-51.