A $60 billion proposal outlined by President Barack Obama on Friday to make community-college tuition free appeared dead on arrival in a Republican-controlled Congress but kicked off a debate about how to usher more Americans through an increasingly expensive and inefficient higher-education system.

Speaking at Pellissippi State Community College in Tennessee during a multistate tour ahead of his State of the Union address on Jan. 20, Mr. Obama framed the plan—named America’s College Promise—as a natural progression of the...