So it was that Mr. Koch found himself standing on the stage of the Metropolitan Opera House nine years ago, in black-tie, being applauded for his generosity to the American Ballet Theater during its annual spring gala.

On either side him: Blaine Trump and Caroline Kennedy Schlossberg.

Whether cultural boards will remain such safe zones for trustees whose fortunes and politics prompt external criticism is very much an open question, given the recent debates over board members who sell opioids or munitions. (Mr. Koch made his fortune in the chemical and fossil fuel industries.)

But on Friday, few who served as trustees alongside Mr. Koch were drawn to that discussion.

Adrienne Arsht, a vice chairwoman of the Lincoln Center board, said she became friends with Mr. Koch when they served on the board of the American Ballet Theater in the 1980s. She recalled attending a performance of “Cabaret” some years ago with him and his wife, Julia Koch, where he sat at a cafe table describing his mother’s role in nurturing his appreciation for the arts. She was a fan, he said, of many art forms, including ballet, classical music and opera.

“He talked so strongly and lovingly about how his mother would take him and his brothers to New York to see theater, and they’d pack in something like five shows in the time they were there,” she recalled.