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From the Definitions of Terms section of the Official Baseball Rules, 2022 Edition:
FIELDER’S CHOICE is the act of a fielder who handles a fair
grounder and, instead of throwing to first base to put out the batter-runner,
throws to another base in an attempt to put out a preceding runner.
So by definition, no throw is made to first base on a Fielder's Choice. It is not necessary for the attempt to put out the preceding runner to succeed in order for the play to be scored FC, i.e., FC does not necessarily mean an out was recorded.
I think Max was on the right track a few posts back.