Does the Run Count on a Tag-Up When the Third Out Is Made at First?

A youth baseball rules question: with runners on 1st and 3rd, a flyout is caught, the runner from 3rd tags and scores, and the runner from 1st is tagged out at first trying to retouch for the third out. The discussion centers on whether that third out is a force out or a timing/appeal play—and whether the run should count.