As a 4th year medical student, gearing up for the match, at a top 20 institution I can more or less assure you guys that networking is much less important than our comrades in business or law. As some other people have said on here going to a top institution can give you access to bigger names that can write you better LORs but knowing a few extra people at other institutions is a drop in the bucket when you're applying to 40+ for residency.
I would definitely say that the name of your school carries a ton of weight. If you look at the residency classes at top institutions (for me, going into orthopaedics), they're predominantly from top medical schools with a few randoms interspersed. There are always exceptions. I also agree that there is a bias at play that top students go to top medical schools go to top residencies. If you are a good test taker and score well on the MCAT you will probably score well on Step 1, etc.
Resources also matter a ton. If you go to a community institution you'd have to claw your way through charts to find any meaningful data to write a paper, for example. In contrast, at top schools you can find yourself paired with MDs who devote a significant chunk of their time to research and pump out papers like a machine. I was able to ride this kind of wave to two first author publications in good journals. I couldn't have done that at a smaller, less well known school.
Anyway, enough of my soapbox.