:"Four percent to five percent of the personal injury cases in the United States go to trial."
Even those cases have the potential of being settled DURING trial at any time up to the moment the jury presents its verdict.
Even after the judgment, the defendant has the right ot appeal if the judgment is believed to be excessive. During an appeal, another round of settlement negotiations could occur.
Example: Plaintiff wins $1,000,000. Defendant appeals on the grounds that the award is excessive. Appeals can take a year or two so the Defendant offers the Plaintiff $500,000 right now and if the Plaintiff accepts, the Defendant withdraws the appeal.
Or, as you wrote, in the absence of sufficient insurance, the Defendant files bankruptcy and any debt above his insurance limit goes away.