finish being done 的用法比较少见,但不能算错误。
在英语原版词典中可以查到类似用法。如:

finish doing sth 属于“及物动词”+“动名词短语”作宾语结构,意思“完成做某事”或者“做完某事”
备注:finish 后面只能接“动名词主动结构”不能接“动名词被动结构”
Tom finished doing his homework.汤姆做完了他的家庭作业。
As soon as he had finished eating, he excused himself.他一吃完就告辞离开了。
Be quiet! He hasn't finished speaking.安静!他还没有讲完。
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