About the distinction in rankings, even if you accept that networking is the biggest thing, recognize that going to a higher ranked school gives you many more networking opportunities. If you need proof, just look at any list of the employers that go to OCI for a T14 vs even a T20 school. Additionally, firms have and want to maintain good relationships with the schools, so there are tons of networking opportunities during the year as well.
Another thing: grade cutoff and school ranking is a real thing. (Source: V10 lawyer flipped through his recruiting binder right in front of me last year.) The higher your school is ranked, the lower your grades can be. Don't assume you can go to a regional or lower ranked school and just "kill it" and be top of your class. In fact, that's a great way to screw yourself.
Unless your connection at a firm is someone who is willing to put their credibility on the line for you if you happen to have poor grade/low ranked school, or maybe your dad is the firm's biggest client, or maybe your dad is senior partner, or maybe you happen to be the most charming person in the world that saved babies in a 3rd world country, you can't count on making buddies with a few alums and getting in that way. As others have said, your best shot is the highest ranked school.
(It isn't that distinct, I don't think, once you can crack the T14, though. I wouldn't fault anyone who takes, for example, a full scholly at NU vs. no aid at Chicago. That said, you can check each school's placement into firm jobs, clerkships, academia, whatever. That's a risk balance you need to decide for yourself.)