“He inherited a very tough, tragic, complicated situation, any way you look at it,” Ms. Farley said in an interview. “His temperament is one of collaboration; he seems to have a low ego need.”

That collaborative approach could help ease tensions at the complex. Although Lincoln Center was founded to serve essentially as the landlord to constituent organizations, over time it began offering its own performances and series, including Mostly Mozart and Great Performers — leading to competition for resources, ticket sales and contributions.

Running the center has never been easy. Reynold Levy, who led it for a dozen years and left in 2014, called his memoir “They Told Me Not to Take That Job.” Things have been rocky since. Mr. Levy’s successor, Jed Bernstein, a producer who had led the Broadway League, was removed after only 27 months. An interim team took over until the center found its next president: Debora L. Spar, who had been the president of Barnard College.

She was pushed out after a year, after clashing with the center’s senior staff, who were backed by the board. Then Lincoln Center named Russell Granet, who had run its education division, acting president — spurring speculation that he was next in line.

Some of the leaders of the complex’s resident arts organizations expressed concerns that Mr. Granet did not have enough prior experience for a job of the scale and stature of running Lincoln Center, according to an arts executive who requested anonymity to discuss sensitive conversations. The leaders wrote Ms. Farley, asking for a meeting. When they met, they discussed the kind of leader they believed the center should seek.