It seems to me that a lot of the debate here is around how each school would help a student with broad interests to narrow down and choose their trajectory. One argument is that MIT offers a more varied array of majors and specialties to explore, whereas the way HMC’s extensive academic core, funneling into a more contained array of majors, could be limiting.
I think the two schools just present students with wide options in different ways. At MIT, where much of what distinguishes the experience from any other university is the ability to take part in top-tier research as an undergrad, the “sorting” of undergrads into UROP jobs has (in my experience, anyway) an outsized influence on the paths students end up in. That is where close relationships with faculty mentors are most commonly built, whereas the classroom experience is less personal, at least in the lower division classes one takes while still making decisions about majors, etc.
This is all YMMV, and there are many exceptions to any generalization. But if I were helping my kid decide between MIT and HMC, I would ask myself whether this student was particularly passionate about exploring their options from a shopping-research-labs perspective, or from a perspective of finding mentors and academic passions in the classroom, and then following those mentors and passions into research opportunities as well as majors and careers.
HMC’s whole philosophy is to keep things broad at the undergrad level, and let students carve out their particular specialty areas within broader programs. This is why the engineering degree is a general one, and why the Core puts every student through their paces across the lab sciences, higher math, and introductory CS and engineering design, before they branch off into majors. (I know MIT has core requirements also, but it’s not the same shared experience that HMC students share as a cohort.) My overall sense is that HMC students approach their sophomore year decision about a major informed by an intentionally-crafted array of experiences, whereas where an MIT student ends up comes from more of a mix of the passions they pursue and the sheer randomness of where particular rolls of the dice (via scheduling, social contacts, EC’s and research gigs) may steer them. I’m not saying that’s bad - life works like that in general! But for a student who potentially has both options, it bears asking whether they’re the kind of person who would best find their path through a more curated, cohort-based educational experience, or whether they would be happier navigating their own path through a dizzying array of choices, knowing that it will take a lot of initiative and decisiveness to get the most out of it. Again, I am not saying that either is objectively better than the other; but a particular individual could be far happier with one experience than the other, depending on their personality, learning style, etc. Both schools accept exceptionally talented and driven students, and produce highly accomplished and successful graduates. The feel of the process is quite different, and a self-aware student is likely to know which sounds more exciting and appealing to them, reputational cachet aside.