The standard claim about Ingersoll is that his unbelief cost him the governorship of Illinois, or a cabinet post, or higher. It is not quite documentation and not quite legend.
What is documented: he was a Republican of real standing, Attorney General of Illinois from 1867 to 1869, and the man who delivered the "Plumed Knight" nominating speech for James G. Blaine at the 1876 national convention, one of the most celebrated pieces of American political oratory of the century, which did not win Blaine the nomination but made Ingersoll's reputation as the finest speaker in the country. He was a valued campaigner for the party thereafter.
What is inference: that the party could not run an outspoken agnostic for office and both sides knew it. Contemporaries said so, he said so himself in his dry way, and no counterfactual can be checked. He was not, in any event, denied an office he had actively campaigned for; he chose the lecture circuit, and it made him wealthy.
The tone is what kept the audiences coming. He was ferocious about doctrine, hell above all, which he attacked as the single most damaging idea ever put into a human mind, and conspicuously gentle about people, and his funeral orations, delivered for friends and for a child of strangers, were reprinted as widely as the polemics.