Graduate schools will hold them in equal regard to the Ivies. Look up Yale and Harvard’s current law school classes by undergraduate colleges represented. You will see these small liberal art colleges excelling. Look up the ‘undergraduate origins of PhDs’ that Reed likes to cite; you will see small liberal arts colleges doing better at producing future PhD’s than almost all the Ivies!
Random people? They will know Princeton and Harvard and Yale but they won’t know Middlebury and Bowdoin. Of course, they also may not know Brown or Dartmouth and will confuse U Penn with Penn State.
Williams grad here. Employers, graduate schools, and people who went to top schools of any sort say, “Wow! Williams.” Distant relatives, gas station attendants, check out counter folks, and other people you may encounter in daily life, for the most part, will ask, “Williams? Do you mean William and Mary? Roger Williams? Where is Williams?”
So, if you want bragging rights with the general populace, famous universities like Harvard and top sports colleges will give you more mileage than our nation’s top small liberal arts colleges. But if your concern is your child’s getting a job or getting into a graduate school, the top small liberal arts colleges will be held in equal regard to the top universities.