Mr. Garrett reflected that uncertainty. Like some of the Trump administration’s more controversial cabinet appointments, he had previously called for abolishing the institution that he would have led. Once nominated, he tried to assure senators that he had had a change of heart. But business and manufacturing groups that benefit from the Export-Import Bank feared that, as the agency’s leader, Mr. Garrett might act to undermine it from within.

They greeted his defeat Tuesday as good news. But at least for the near term, the bank’s future remains unresolved. Since 2015, the agency has been hobbled by a lack of personnel necessary to approve projects over $10 million, formerly the bulk of the agency’s work. An estimated $42.2 billion worth of deals are stuck in the pipeline waiting for approval, which could support an estimated 250,000 American jobs, a spokeswoman for the Export-Import Bank said.

The Export-Import Bank has offered government-backed loan guarantees to overseas customers wishing to buy products manufactured in the United States since the Depression. Some of its biggest customers are General Electric, Boeing and Caterpillar, whose equipment can be out of reach to some foreign customers without loan guarantees that help them secure credit with private banks.

But the agency — once backed by large majorities in both parties — has become something of a litmus test for hard-line conservatives versus establishment Republicans. Hard-line conservative outside groups like Heritage Action and the Club for Growth have demanded the agency’s closing. Business groups like the National Association of Manufacturers and the U.S. Chamber of Commerce have tried to revive it.

As a candidate, Mr. Trump described the bank as a kind of corporate welfare. But in his first few months in office, the president appeared to have been persuaded that the bank’s mission dovetailed with his own goal of using the might of the government to help lift manufacturing and exports. He also appeared to have been persuaded by the bank’s supporters, who point out that the agency makes enough money to more than pay for itself and that other countries have far larger export credit agencies.