If it’s going to give your kid a chip on the shoulder, then by all means, do what you need to do to get into the better schools
No, don't. Your son is already in an excellent school. He'll get a fine education at Colgate (assuming he studies) and when he emerges and proceeds with his career will be (correctly) perceived as having gone to an elite college. The annual WSJ college survey asks recent alums if they think their alma mater (1) was "the right choice," (2) had an "inspiring" student body, and (3) was "worth the cost." Colgate's recent alums not only rank it highly (8.6, 8.2 and 8.6 of 10 respectively), but on all three measures, Colgate grads give their school higher rankings than grads of Penn, Columbia, Cornell, Northwestern, Hopkins, Amherst, and Hamilton give theirs. Again, on all three measures. People who go to Colgate generally like it, if anything more than people who attend comparable (what PP calls "better") schools like theirs.
Your kid is, to be frank, having an exaggerated, silly and immature reaction to a very minor "setback," and attaching a wholly unwarranted significance to what are actually relatively minor distinctions among top ranked schools and the students who attend them (seriously, in the real world - outside of college applicants and their hysterical parents on DCUM -- how many people find the quality of individuals is stratified by the college ranking of their alma mater?). Freshman get to their chosen college having a whole lot of preconceived notions about what that school will be like, and whether it's a good 'fit' - then after a few months of freshman fall partying, the actual lived experience of classes and activities and friendships and off-campus life take over. You'd be doing the most to help your son deal with this "situation" (

) and grow up if you pushed/compelled him to attend Colgate and agreed to let him decide only after a year (not say the first six weeks) if he wants to transfer to a different school that offers something Colgate doesn't and accepts him.
And ignore the periodic undeserved trashing of Colgate on this site -- it's mostly from people who aren't familiar with Colgate (you'll note most of the critiques are either second hand or from someone who once briefly visited the campus). You don't have to scratch too deep to realize some of this is probably from people who feel they or their kid wouldn't fit in with Colgate's happy, well-adjusted student body.