The issue is not any throw to first vs only a throw to retire a B/R, it's that you don't have a batter-runner anymore after the fly ball is caught.
The running lane is only applicable to the batter-runner. Once the fly ball is caught, you have a retired runner and that runner must commit an act of interference in order to interfere. I would say that simply continuing to run toward first base does not qualify, especially with the play and the ball behind him/her.
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