This 100x.
By the time you're getting your PhD, the college graduate should have read the classics, done at least 1st year level calculus, biology, physics, ethics, philosophy, and 2-4 years of a foreign language.
High school and undergraduate is the time to be broad-minded. PhD is for very focused work (with the big picture in mind).
Personally, I read the classics and learned Greek and Latin (high school), then in college took 8 courses of philosophy, 2 of Spanish, 2 in economics, 4 in business/entrepreneurship, and a smattering of other interesting things (art, music, etc...) while getting a Physics major.
Then I went and spent 6 years learning more about one thing than anyone in the world.
It may be more productive to have the expectation of bigger-thinking introduced earlier and let that expectation be realized more fully at the PhD level.