It was early 2021, and Senator Rick Scott wanted to go big. The new chairman of the Senate Republican campaign arm had a mind to modernize the place. One of his first decisions was to overhaul how the group raised money online.

Mr. Scott installed a new digital team, spearheaded by Trump veterans, and greenlit an enormous wave of spending on digital ads, not to promote candidates but to discover more small contributors. Soon, the committee was smashing fund-raising records. By the summer of 2021, Mr. Scott was boasting about “historic investments in digital fund-raising that are already paying dividends.”