It was chicken tenders night at Yale’s chapter of Hillel, the Jewish student group, and the basement dining hall was packed with boisterous, hungry students attracted by overflowing vats of kosher fried chicken and mac and cheese.

Some students kissed the mezuza on the way in. Others were not even Jewish, but came for the food and companionship, a sign of the pluralism that Hillel — the dominant Jewish campus organization in the United States — says it embraces.

Yet under the surface,…