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If I am applying to big law firms through the website (I don't have any connections and the firm doesn't accept applications from recruiters), how important is the cover letter? Does it really help more than hinder?

It probably won't hurt anything, but it's unlikely to be read. If you have a specific connection to someone at the firm, reach out directly rather than name dropping them in the cover letter. In general, cover letters are being phased out in job applications. They really only ever made sense in the context of paper applications.