“I don’t think that he sees the role of the chief of staff to alter the personality of the person he works for,” said former Representative Trey Gowdy, a friend and fellow South Carolina Republican who golfed with the president and Mr. Mulvaney last month. “They have a good relationship, but the president’s the boss.”

Some outsiders see the cascade of hard-line policy ventures, unorthodox appointments and high-level purges of recent days as a sign of Mr. Mulvaney’s expanding influence, assuming that he is pushing Mr. Trump to the right. But insiders call that a misconception, insisting that Mr. Mulvaney at most is pushing on an open door and otherwise is merely liberating Mr. Trump to pursue the courses he prefers.

The lesson Mr. Mulvaney took from the unhappy experiences of Mr. Priebus and Mr. Kelly — both of whom were cast out unceremoniously via Twitter or comments to reporters — was that Mr. Trump is not interested in being managed by aides who think they know better, and so he has tried to build a process that he thinks better serves the president.

To some Republicans who had become resigned to an ungovernable White House, Mr. Mulvaney has been a welcome presence, one who has tamped down some of the tribal rivalries that played out in vicious form through clandestine disclosures to the news media.

“I think he’s become an asset because he’s learned how to keep the Trump train running,” said Scott Reed, the top political adviser to the United States Chamber of Commerce. “A combination of managing the staff, raising the morale and limiting the leaking has made him a successful Trump chief of staff.”