BLOOMFIELD TOWNSHIP, Mich.—Paul Azinger revamped the Ryder Cup qualifying system to get the best players, and he had no complaints with the eight Americans who earned a spot on the team.

But not even four captain’s picks to follow will change the way he sees this competition against Europe.

“We are going to take on an underdog role in this Ryder Cup for the first time in a long time—even on paper—when you look at the strength of the European squad,” Azinger said Monday morning after the PGA Championship. “We are going to have everything to gain here.”

That’s...

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