It's a misunderstanding that CMU focuses on applied statistics; Check out Wasserman, Ramdas, or Balakrishnan. Actually, except for Barber, who is one of the best theorists in the world without a doubt, I think nobody can match them in terms of theoretical research and advising. Gao and Ma are amazing, but they are junior faculties and have not proven their ability for advising PhD's. In fact, except for 6-8 Professors, many of faculty members in UChicago can be viewed as biostatisticians. However, UChicago has another statistics department in the Booth School of business, which has strong theorists such as Rockova. I see some recent graduates were advised by those Booth faculty members, wrote JRSSB or JASA and went to prestigious postdocs (although I do not know how much opportunities stat PhDs in physical sciences division have with Booth). Therefore, for theoretical research, I think CMU and UChicago provide similar opportunities. I think if you can work with Barber, UChicago is slightly better, but she only accepts one PhD student a year.
UChicago is a very prestigious school as an undergrad institution. They are excellent in almost every fields they deal with, except for computer science. They do not have engineering department, and their CS department is ranked below 50. CMU is ranked 20-ish as an undergrad institution, but they are top notch in computer science and machine learning, and many of their research are related to statistics department and many of their faculty members advise stat PhD students; check out Aarti Singh's publications and students, for example. In my opinion, the prestige as an undergraduate institution is only applicable to undergraduates, and PhDs would only enjoy the prestige of their graduate program. In terms of statistics, I think UChicago has a slightly more reputation than CMU. For machine learning, however, UChicago is not even in top 30.
UChicago offers very rigorous coursework and quals, which are very nice but is an overkill in my opinion. The quals forbid students from doing research in their first years. I believe that CMU's courseworks provide the key skills needed for the research of the theory of modern statistics. You can check out lecture notes and also Youtube videos of CMU 36-705 or 36-702. Also checkout this syllabus (https://www.stat.cmu.edu/~aramdas/martingales18/m18.html). You will figure out the style of CMU's courses. To sum up, I think CMU provides necessary skills, while UChicago trains you more than enough. Either way is good, and it's your choice which training you will have.
In terms of placements, surprisingly, UChicago is leaned toward industry; check out this page (https://stat.uchicago.edu/alumni/phd-alumni/). The cohort size is around 15-ish, and the ones not listed there are probably in the industry. Note that the listings include non-stat PhDs, and also they are not initial placements; Walter Dempsey, for example, is listed as an Assistant Professor at UMich biostat, but he did Postdoc for four years. In my opinion, Yale and UW are better than UChicago when it comes to producing assistant professors doing theoretical research. The postdoc placements of UChicago are amazing, though.
