A few points for starters...
There's no rule that says play is automatically dead if a stray ball comes in from another field. There's nothing in the rule book about "ball interference". However...umpires do have the discretion to call time as they see fit and to correct any unusual situation that falls outside of the rules.
We routinely call time if a ball comes on the field before a pitch or play. This prevents the remote chance of the stray ball even possibly interfering. This is more of a preventive measure, to head off any problem before it has a chance to happen.
If a ball enters the field in the middle of a play, you should keep the ball live and let the play continue. It's a lot easier to judge what "might have happened" if you let it play out, as opposed to killing the play and making a guess.
Let me throw out an example. Suppose the batter hits the ball to the outfield fence, then a loose ball rolls through the infield. Are you going to deprive the batter of an extra base hit and rule a "do-over" when the stray ball had zero impact on the play? I hope not!
On the other end of the scale...suppose there's a sharply hit ball right to the shortstop, than another ball shoots onto the field right at her. With two balls flying right at her, you could say that the other ball impacted the play.
There could be endless variations and possibilities in between those two extremes. Each would need to be judged on it's own merit, so there's no "one size fits all" ruling that an umpire might make to correct things.
On your play...with a slow roller to the shortstop, how in the world did the first baseman think that another ball coming her way, on the opposite side of the field and from a different direction, was the game ball? Why would she abandon the play at hand to chase down a loose ball that obviously wasn't the game ball? If she just assumed that the play was dead and gave up on it...that's on her.
I would say leave the ball live and let it play out. Players have the responsibility to know what's going on around them and to act accordingly. I'm having a hard time picturing how this loose ball might have had any real effect on the actual play or why F3 would go after the other ball.
I'm also sure that the defensive coach will probably argue for some kind of "interference" if the batter safely reaches first base, but there isn't really any rule based justification for that argument.
