Three years ago, when Amy Coney Barrett was a nominee to the federal appeals court in Chicago, she endured a blistering Senate confirmation hearing in which Democrats grilled her on whether her deep Catholic faith would distort her view of the law.

She maintained a calm demeanour under the intense questioning, but behind the scenes, she was “a little shook,” recalled Laurence Silberman, a Washington federal appeals court judge for whom Barrett once clerked and who sat with her family at the 2017 hearing...