Yes connections can help you get a job. But I don’t think you are correct that going to columbia college will be much help over BMG if you have similar grades, connections, and presentation.
I think you are too pessimistic regarding getting biglaw from elite schools. That isn’t to say it is easy, but if you can get good grades from an elite school you have quite a good likelihood.
Also, yeshiva guys do have good connections. Often much better than their classmates