BUENOS AIRES — Seventy-five team directors had placed their votes in envelopes, and Argentine soccer was moments from electing its first new leader after 35 years under the rule of Julio Grondona, a strongman of Latin American soccer politics who died in 2014. But hope quickly turned to despair: The 75 votes somehow produced a 38-38 tie, a stunning rejection of basic mathematics.

The botched election for the presidency of the Argentine soccer federation this month was viewed by many Argentines as a national disgrace. But in its dysfunction, it offered a view into the perils faced by the new figures battling to control the region’s influential soccer organizations in the wake of a sweeping corruption scandal, and how difficult it will be to coax leaders whom fans can trust out of what many consider a tainted talent pool.

“I can’t trust any of them now — they’re all sullied,” said Patricia Rodríguez, 55, a travel agent, referring to Luis Segura and Marcelo Tinelli, the two candidates in the tied vote. “It’s such a shame, because I had great hopes.”

Ms. Rodriguez’s comments could be repeated all over South America. Of the more than 40 defendants charged in the United States Justice Department’s broad investigation of corruption in world soccer, more than half hail from South America. Every country in the continent’s 10-member confederation, Conmebol, has had at least one senior soccer official charged in the case. The defendants include the past three Conmebol presidents, three former presidents of the Brazilian national federation and the current or former presidents of the federations of Bolivia, Chile, Colombia, Ecuador, Peru and Venezuela.