This is the wrong way to choose a recommender.
This, on the other hand, is a step in the right direction. Which recommender knows you best? Who’s seen a side of you that goes beyond the classroom? Who can speak not just to your academic credentials - which all competitive applicants will have - but also to the things that set you apart from the rest of the applicant pool?
As long as he/she can write effectively - and there are high school dropouts who’ve mastered the English language in a way some Yale graduates (including a recent US president) never will - it doesn’t matter what your teacher’s alma mater is. A Harvard graduate who just knows that you’re a bright kid will write a letter that isn’t nearly as compelling as that of a University of Alabama graduate who can comment on your character and the traits that make you different from John or Jane Doe. There are students who’ve been rejected because they had letters of recommendation from congressmen or senators - who barely knew them, couldn’t say much, and ended up writing far worse letters than the applicants’ teachers would’ve produced.
With that out of the way, I have to say that with a GPA of 3.5 (at most) by application time, you shouldn’t fall in love with UChicago just yet. The U of C has to turn down more than 90% of applicants, and competition for a small number of spaces is stiff even among straight-A students.