Perhaps some of the reasons that people seek elite private colleges:
a. The perception that public and non-elite private colleges are inferior.
b. Having a goal of some elitist profession (e.g. Wall Street or management consulting) that preferentially recruits at elite private colleges.
c. Perceiving that some professional goal requires an elite college (even if it does not).
d. A top student in a high school being told over time by teachers, parents, etc. how s/he can aim higher than the state flagship, internalizing the idea that the state flagship (and similarly selective other private and public schools) is “beneath” him/her.
Of course, realism means that, if one does not have a large unearned advantage (e.g. parent who donates a building to the college) and is not so greatly talented that high but not excessively stressful levels of effort produces top end academic credentials plus top end other achievements (particularly recruitable athleticism), then getting into an elite college is the longest of long shots, and excessively stressful levels of effort may result in a bigger letdown when the rejection letters come, as well as other negative effects on one’s life.