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The official scorer isn't allowed that. The rule states if the catcher makes an errant throw there is no error and the runner gets a stolen base. The error comes from the runner being able to advance. No assumption of an out. The log shows it was a throwing error.
If the error was because the fielder muffed the catch then yes the runs would have been unearned. The rules further states in that case the runner is charged with a caught stealing and the fielder throwing the ball an assist even though the runner was safe.
The scorer could have assumed an unearned run if the next batter got an infield hit, scoring a run, followed by an out; because the run would not have scored without the baserunner advancing on the bad throw.
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