Is the enforcement of hudud punishments restricted to the Imam or his deputy, or is it permissible for individuals to apply them in certain cases, based on hadiths such as the story of Sa'd with his wife, the woman who insulted the Prophet, and 'Umar's statement to the one who killed a man for committing adultery with his wife?

Your statement that you do not want the words of scholars is wrong, because their words are an explanation of the words of Allah and His Messenger, and they are the ones who distinguish between the clear (muḥkam) and the ambiguous (mutashābih). All that you have mentioned contains no proof for what you want to establish. And the Prophet, peace and blessings be upon him, did not approve of Sa’d’s statement, rather he clarified the intensity of his jealousy. The ruling on the matter is known, and it was stated by the man who asked the Prophet, peace and blessings be upon him. And the woman you mentioned who was killed was the Umm Walad (concubine who bore a child) of Ibn Taymiyyah, not his wife, and she was killed because she was insulting the Prophet, peace and blessings be upon him, and this was in the early days of Islam, and there is no proof in it for what you want. The story of Umar and the man is not proof, because the man defended his womenfolk, he did not carry out a prescribed punishment (ḥadd). And if it were an argument, Ali would not have said: "If he does not bring four witnesses, then he should be given up entirely." And Al-Bukhari titled a chapter, "Chapter: One who saw a man with his wife and killed him." And the majority opinion is for retribution (qiṣāṣ). Ahmad and Ishaq said that his blood is not avenged if he establishes proof. Al-Shafi'i said that it is permissible for him to kill her in what is between him and Allah if she was a previously married woman, but retribution (qawd) is not waived for him in the apparent ruling. And giving every person the right to exact the prescribed punishment himself corrupts the situation and leads to false claims. And the purification of witnesses cannot be undertaken by every person. So, the execution of prescribed punishments is the prerogative of the Imam or his deputy, and anything else corrupts affairs, and the texts of the Sharia and its general rules indicate this.