U.S. influencers and an administration official arrived in Russia just as Vladimir Putin, the country’s president, needed a publicity boost. The small group of Americans flew in as the St. Petersburg International Economic Forum, or SPIEF, often dubbed Russian Davos, opened on June 3 in the port city. The annual conference serves as a promotional vehicle for the country as Putin seeks to project strength and global power. Officials say this year they are receiving 20,000 guests from over 100 countries, with 76 countries sending high-level representatives. Putin has increasingly faced setbacks on the battlefield as a Ukrainian drone strike...