Chicago does have an engineering major, albeit only one: molecular engineering. It was started a few years ago. They even started a new school of molecular engineering. But it does not have any of the traditional engineering fields like mechanical, electrical, etc. The physics department is quite strong, but heavily focused on theoretical aspects and less so on applied.
I’m familiar with the administration of both schools, and I’d say that UChicago has an advantage. They listen to the students and parents. Not sure I feel the same about Columbia where they are often accused of being tone deaf. Free speech is pretty similar at both schools. Columbia is very tolerant. Have you heard about the mattress girl?
Some notable issues about Columbia: Lots of students get waitlisted for their course selections, and don’t often get their first choice until upperclassmen. The core differs between SEAS and CC. SEAS students have a higher credit requirement for graduation. Its also very difficult to double-major. Proximity to NYC firms makes it possible to do an internship during the school year. Cost of living is higher in NYC.
UChicago considerations: quarter system makes for a very fast pace. About a 1/3 of your classes will be core, 1/3 major and 1/3 electives. Easier to double major at Chicago. Finishing classes in early June makes it difficult for some summer internships (where start dates of May is common). Not impossible but just keep in mind. Winters are brutal in Chicago. Also keep in mind that south of the Plaisance there is occasional violence.
I agree with above posters who suggest combining EA/ED strategically. You could theoretically do ED1 Columbia + EA Chicago. Then if you get deferred switch the EA app to ED2. This would give you 3 shots on goal in the early round.