Not exactly if one actually looks at the history.
US colleges didn’t become the best in the world until during/after WWII due to the combination of massive brain drain from Fascist occupied countries in Europe and Asia from the 30’s and 40’s, massive US government funding of science and area studies programs(Roots of Eastern European and East Asian studies programs in Elite/respectable colleges), massive US government spending on hundreds of college campuses to send officer trainees for some university education/exposure(i.e. US Navy V-12 program), and of course…an influx of returning WWII veterans taking advantage of the GI Bill to enroll in hundreds of colleges across the country.
Before WWII, the best universities in the world were in Europe.
Some of the best research universities before the early '30s were in Germany. However, once the Nazis took over, their anti-semitic and intolerance for perceived political opponents meant most of the topflight Profs were forced to leave their jobs and emigrate. Most of them ended up here which benefited the US colleges system at the expense of the German universities where they were formerly employed. To a large extent, the German University system and the European universities to a lesser extent are still making up for the cumulative effects of Nazi policies/war/occupation to this very day.
A glimpse of this could be seen from this American Math Society article about how Gottingen’s status as a topflight institution in the world for the study of math precipitously declined once the Nazis came into power and imposed their antisemitism and intolerance for even a whiff of political dissent:
http://www.ams.org/notices/199510/maclane.pdf
Actually, in most other parts of the world, including societies with higher performing K-12 students, parents aren’t allowed to choose which public school to send their kids and up until recently…parents mostly did defer to the expertise of their educators. However, the academic standards by the centralized board of education for the public K-12 tended to be set very high.
One illustration of this was an older HS alum who left the ROC(Taiwan) after attending finishing 5th grade at a regular public elementary school in a working-class neighborhood. Despite being placed in the SP/gifted classes in US public elementary and middle schools, he recounted he literally learned nothing new in math or most other subjects except English and US history/social studies until well into second semester of 9th grade at our STEM-centered public magnet.
And his older brother who graduated from our HS a couple of years earlier than him had the exact same experience…except he coasted until his junior year of HS because he had a year or two of middle school which included lab sciences most US students take in HS(biology/chem/physics with lab).