You’re not understanding this cliche correctly. Ivy League universities don’t want well-rounded students they want well rounded student bodies.
For a long period in the ‘80s and ‘90s there was a common refrain that said students needed to be ‘well rounded.’ This may have been accurate at the time, and might still be for less selective schools, but it is useless for the purposes of Ivy League admission.
Ivy League admissions officers think in terms of well-rounded student bodies not well-rounded students. That is, they would rather have a class of award-winning poets, math prodigies, and polyglots than they would a class of students who all took the same classes, participated in the same clubs, volunteered at the same organizations, and played the same instruments.
We wrote a long-form article about this topic, but the relevant graphic is below. Imagine a school wants to have the most green space covered in its student body (far right) so that they can have the most outstanding school.
You can either choose ‘well rounded’ students like students F, G, and H, and you would end up with a student body that’s missing outperformance.
Or, you can choose three complementary outperformers, like students A, B, and C. In this case, your student body would be outstanding across the board, making it a better ranked, more competitive school.
So try to be well rounded in that you have competitive grades and test scores, but try to be exceptional in one other thing rather than mediocre in a few things.