Is my family's prayer valid after I stopped leading them in prayer, claiming that the ablution was invalid, with my prior intention to repeat the prayer as a supererogatory or obligatory one after performing it individually?

Your second prayer is a supererogatory one, and it is permissible for someone performing a fard (obligatory) prayer to follow someone performing a nafila (supererogatory) prayer. This is the view of the Shafi'i school and a position held by some Hanbalis. They based their argument on the fact that Mu'adh ibn Jabal used to pray with the Messenger of Allah, peace and blessings be upon him, and then lead his people in prayer, with his prayer being supererogatory for him and obligatory for them.

If the prayer of the Imam becomes invalid, the prayer of the followers does not become invalid; rather, they should appoint someone to complete the prayer with them. This is based on what Umar ibn al-Khattab did when he was stabbed: he put forward Abdur-Rahman ibn Awf, who then completed the prayer with the people. This occurred in the presence of the Companions, and none of them objected.