One reason why the United States allows very broad discovery is because courts impose generally lax pleading requirements.

A case typically starts with the plaintiff’s Complaint.  In the Complaint, a plaintiff usually does not need to include the evidence supporting her claim.  In the federal courts, she usually only needs to include in the Complaint sufficient factual allegations that a reasonable person would conclude that her claim against the defendant is at least plausible.   The defendant’s Answer, Affirmative Defenses, and Counterclaims will also likely have little, if any, evidence.   

But eventually the parties need to prove their cases.  At some point a jury or judge will need to decide whether one side has proven its claims.

Discovery “cures” the shortage of facts and information at the beginning of the case.  Through the discovery process, the parties obtain and share evidence and information.    The United States generally favors broad discovery so that a judge or jury can rationally decide the case on a complete set of evidence.  Below is a video I uploaded a while back on discovery and pleading requirements.