Republicans in Congress, who have vehemently opposed the agency since its creation, have also been unable to muster enough support to derail its work. Efforts to strike down a rule ordering new consumer protections on prepaid debit cards never made it to a vote in either the House or the Senate.

“The public does not share the G.O.P.’s ire toward the agency or its mission,” said Dean Clancy, a Tea Party activist who worked in the White House under President George W. Bush and is now a policy analyst who tracks actions of the consumer bureau. “It is an agency about protecting the little guy, and that is tough to oppose.”

The stories of gratitude rounded up by the agency’s staff for Mr. Cordray illustrated its appeal. Among them was a homeowner in Tennessee who got a disputed lien removed from a property, someone in Kentucky who got assistance warding off a debt collector pursuing a medical bill that had been paid, and a person in Pennsylvania who said the agency helped resolve a contested credit card debt.

That doesn’t mean the Trump administration and other opponents have given up on neutralizing the bureau’s work.

Administration officials have isolated the bureau from parts of the government that, under President Barack Obama, helped fulfill its mission. In public statements and documents, officials at the Justice Department, the Treasury Department and the Office of the Comptroller of the Currency have all turned a cold shoulder toward Mr. Cordray and his staff.

Lobbyists for the financial industry are working behind the scenes on efforts to dismantle some of the bureau’s signature initiatives, according to people directly involved in the plans. They include lawsuits to be filed in reliably conservative courts when new regulations are issued.

For now, though, it is mostly a waiting game. Mr. Cordray’s term as director expires next July, when he could be replaced with a sympathetic Trump appointee. That moment could come earlier as there is speculation that Mr. Cordray might resign — perhaps soon — to enter the Democratic primary for governor in Ohio.