WASHINGTON—While America awaited the special counsel’s Russia report, the man who will make key decisions about its disclosure to the public was quietly pursuing other objectives at the Justice Department, knowing his legacy goes beyond how he handles the high-profile investigation.

Robert Mueller’s findings were delivered on Friday to Attorney General William Barr, who was confirmed to his post last month. Before then, he had been focused mostly on everything else: getting briefings from Federal Bureau of Investigation officials…