Also, surely the interview will ask you about your background knowledge of economics, and if you didn't know who Keynes was you're unlikely to get an offer.
In the US, a bachelors degree takes 4 years, consists of extensive 'general education' requirements outside of the emphasis area, and generally start with intro courses that are more in depth than what would have been studied in high school. So a background in the particular area is not required.
Knowledge of specific economists is not the norm in the basic college prep curriculum in the US. Students in the IB program (relatively rare, available in major cities so the majority don't have the option) or the AP program will have studied the subjects to a greater depth. At least equivalent to freshman/sophomore college courses in the subject.
For a bachelors degree the students will apply to schools and get in based on: GPA, SAT/ACT scores, extracurricular activities, interview (not used for the average university), and essays. Some schools put all freshmen into the same pot and require them to apply to their major after their freshman year. In that case, it'd be similar to your "Don't know Keynes? Probably won't get an offer." situation. However, that's also not the norm. And once at the school the students can usually change major with ease.
"The Faculty does not have standard requirements for subjects other than Maths: in particular A level Economics, though useful, is not a prerequisite."
http://www.econ.cam.ac.uk/ba/guide.pdf
(FWIW I did mathematics as an undergrad but I considered CS and spoke to CS faculty at various university open days, from what I remember they all considered maths to be the only essential A level for admission.)