It depends what kinds of teens you’re dealing with.
Typically, the most “prestige-obsessed” are those for whom it matters the least.
Research shows that attending a top college (top 50 or top 25, especially) makes a big difference for specific types of teenagers: lower income, first generation, under-represented minority. Because they don’t have a family circle or professional support network, their college’s “brand” makes up for it, and the college has enough resources to provide what they need (academically, financially, etc.)
For academically strong teenagers who are not Ivy-material but are First-gen, lower income, or URM, starting at a 4-year college also matters. (I saw a study where students from these groups and (would) rank in the top 25% for their flagship but start at a CC have lower odds than bottom 25% students who start at the flagship.)
For middle class students, students whose parents attended college… what matters is the student’s attitude, resilience, ability to question themselves, to seek help, to make the right choices, etc., much more than the college.
There are some exceptions: Investment Banking recruits from very specific colleges; Acting, musical theater, music, anything in the entertainment industry is peculiar and the program itself matters (not the university); teaching requires a college with good connections to local schools (and if you want to teach at a Catholic school, it may be better to attend a Catholic college, or attending a regionally prestigious private college is important if you want to teach at a prep school); International Relations is another peculiar one, where high standards in foreign language, strong reputation in political science or IR, and strong study abroad are necessary; a student who’s advanced in math (more than calculus BC senior year) needs to check math offerings at the college, since not all offer sufficient classes, and the more advanced the student is (ie., if the student was taking linear algebra junior year in high school…) the more important it is to check the offerings. Same thing for students who are strong in a foreign language - if they reached B1 (AP) or spent a HS year abroad and are fluent, check that several upper-level classes are offered each semester, and not just the 101-102, 201-202 sequence + advanced grammar, check what level is expected for majors (often, B2… so that if your student is already at B2 as a high school senior, it’d be a terrible fit!) For students who want Accounting, checking which firms recruit on campus is important, as is the fact the college offers an accredited 5th year OR has tracked where the seniors went for their 5th year and can provide information about their situation. For some majors that seem a bit “less common” than most, such as Art History or Asian Studies, it’s essential to attend a university that has a strong, academically-recognized and academically- connected program - ie., Art History majors at Williams vs. Art History at Bowling Green State University do not have the same “doors” at all, to the point they’re not even knocking on doors on the same planet.
On the other hand, any flagship, LAC ranked up to 125, and regional university up to 20 or so, will provide a decent premed program - what will distinguish the students’ success is the overall support at the university (are the premed classes weed out? what’s tutoring like? how well do professors know the process? Are they connected to any med school?) What are the opportunities for shadowing, for experience at a clinic, for research? The university’s ranking doesn’t matter, and many premeds will find that attending a university ranked #52 will be better for med school odds than attending a university ranked #12. (In the example given upthread, about attending a small college over Yale, it was in fact a Top 100 college, Rhodes, that is well-known for its excellence in the natural sciences and the support it gives its premed students.)
For engineering, as long as the program is ABET-accredited, it’s fine - classes will be rigorous, classmates will be motivated, and outcomes will be as good as the economy makes them.
Note: Parents who obsess about colleges don’t obsess about Top 100. It’d be too easy. 
They obsess about the exact same few schools - Harvard, Yale, Princeton, Stanford, MIT, UChicago, Penn, Williams, Amherst, UVA, UMichigan, UC Berkeley, UCLA, more rarely Cornell, Brown, Pomona…