WASHINGTON — The Senate voted on Wednesday to roll back President Biden’s vaccine mandate for health care workers at federally funded facilities, in a symbolic move orchestrated by Republicans who are pushing to weaponize pandemic precautions against Democrats in this year’s midterm congressional elections.

Almost two years after the director of the Centers for Disease Control and Prevention called for 100,000 contact tracers to contain the coronavirus, the C.D.C. said this week that it no longer recommends universal case investigation and contact tracing. Instead it encourages health departments to focus those practices on high-risk settings.