I don't follow the logic the author is using. Let's say we make all public colleges free for students. "Elite" schools will still exist, and top students will still target schools such as Ivy/MIT/Stanford/etc. One look at the people in executive offices or top political offices and you'll find a substantial portion are graduates from elite schools.
The article argues that public colleges being free "[...] will alleviate the student debt crisis and enable low-income students, particularly those of color, to go to college."
Your point about elite schools still existing can (and will) be true at the same time in this hypothetical universe where US public college tuition is free.