Isn't the noble Prophetic hadith, in which feeding one's family is mentioned as part of expiation, evidence for the permissibility of giving expiation to a single needy person? And consequently, isn't every ijtihad that contradicts this text void?

The aforementioned hadith is agreed upon and does not indicate what you mentioned. The Prophet, peace and blessings be upon him, gave the dates to the man to feed his family as charity due to his need, not as expiation. Scholars have differed regarding the waiving of expiation due to indigence: some said it is waived if the indigence is concurrent with its obligation, because he was not commanded to pay it if he became wealthy. Others said it is not waived, and the expiation that was permitted was not in the nature of expiation. Then they differed on whether it was specific to that man or abrogated. It was also said that since he was unable to provide for his family, it was permissible for him to spend the expiation on them. The stronger opinion is that he gave him the dates as charity for him and his family due to their need, not as expiation. The expiation was not waived but remained a debt upon him.