SEOUL — President Biden embarked Thursday on his first diplomatic mission to Asia since taking office, hoping to demonstrate that the United States remained focused on countering China, even as his administration stage-managed a war against Russia in Europe. With his original strategy of pivoting foreign policy attention to Asia effectively blown up by Russia’s invasion of Ukraine, Mr. Biden has now shifted to the argument that there can be no trade-off between Europe and Asia and that only the United States can bring together the democracies of the East…